geometry problems from International Mathematical Olympiads Shortlist
2015 IMO Shortlist G1
Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ =AH$.
2015 IMO Shortlist G2 (HEL) problem 4
Let $ABC$ be a triangle inscribed into a circle $\Omega$ with center $O$. A circle $\Gamma$ with center $A$ meets the side $BC$ at points $D$ and $E$ such that $D$ lies between $B$ and $E$. Moreover, let $F$ and $G$ be the common points of $\Gamma$ and $\Omega$. We assume that $F$ lies on the arc $AB$ of $\Omega$ not containing $C$, and $G$ lies on the arc $AC$ of $\Omega$ not containing $B$. The circumcircles of the triangles $BDF$ and $CEG$ meet the sides $AB$ and $AC$ again at $K$ and $L$, respectively. Suppose that the lines $FK$ and $GL$ are distinct and intersect at $X$. Prove that the points $A, X$, and $O$ are collinear.
Let $ABC$ be a triangle with $\angle C = 90^o$, and let $H$ be the foot of the altitude from $C.$ A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.
2015 IMO Shortlist G4
Let $ABC$ be an acute triangle, and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ again at $P$ and $Q$, respectively. Let $T$ be the point such that the quadrilateral $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of the triangle $ABC$. Determine all possible values of $BT / BM$.
2015 IMO Shortlist G5 (SLV)
Let $ABC$ be a triangle with $CA \ne CB$. Let $D, F$, and $G$ be the midpoints of the sides $AB, AC$, and $BC$, respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I' $ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$.
2015 IMO Shortlist G6 (UKR) problem 3
Let $ABC$ be an acute triangle with $AB >AC$, and let $\Gamma$ be its circumcircle. Let $H, M$, and $F$ be the orthocenter of the triangle, the midpoint of $BC$, and the foot of the altitude from $A$, respectively. Let $Q$ and $K$ be the two points on $\Gamma$ that satisfy $ \angle AQH= 90^o$ and $\angle QKH= 90^o$. Prove that the circumcircles of the triangles $KQH$ and $KFM$ are tangent to each other.
2015 IMO Shortlist G7 (BUL)
Let $ABCD$ be a convex quadrilateral, and let $P, Q, R$, and $S$ be points on the sides $AB, BC, CD$, and $DA$, respectively. Let the line segments $PR$ and $QS$ meet at $O$. Suppose that each of the quadrilaterals $APOS, BQOP, CROQ$, and $DSOR$ has an incircle. Prove that the lines $AC, PQ$, and $RS$ are either concurrent or parallel to each other.
A triangulation of a convex polygon $\Pi$ is a partitioning of $\Pi$ into triangles by diagonals having no common points other than the vertices of the polygon. We say that a triangulation is a Thaiangulation if all triangles in it have the same area.
Prove that any two different Thaiangulations of a convex polygon $\Pi$ differ by exactly two triangles. (In other words, prove that it is possible to replace one pair of triangles in the first Thaiangulation with a different pair of triangles so as to obtain the second Thaiangulation.)
2016 IMO Shortlist G2 (TWN)
Let $ABC$ be a triangle with circumcircle $ \Gamma$ and incentre $I$. Let $M$ be the midpoint of side $BC$. Denote by $D$ the foot of perpendicular from $I$ to side $BC$. The line through $I$ perpendicular to $AI$ meets sides $AB$ and $AC$ at $F$ and $E$ respectively. Suppose the circumcircle of triangle $AEF$ intersects $ \Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $ \Gamma$ .
Let $B= (-1,0)$ and $C = (1, 0)$ be fixed points on the coordinate plane. A nonempty, bounded subset $S$ of the plane is said to be nice if
(i) there is a point $T$ in $S$ such that for every point $Q$ in $S$, the segment $TQ $ lies entirely in $S$; and
(ii) for any triangle $P_1P_2P_3$, there exists a unique point $A$ in $S$ and a permutation $\sigma$ of the indices $\{1, 2, 3\}$ for which triangles $ABC$ and $P_{\sigma (1)} P_{\sigma (2)}P_{\sigma (3)}$ are similar.
Prove that there exist two distinct nice subsets $S$ and $S' $ of the set $\{(x; y) : x \ge 0, y \ge 0 \}$ such that if $A \in S$ and $A' \in S' $ are the unique choices of points in (ii), then the product $BA \cdot BA'$ is a constant independent of the triangle $P_1P_2P_3$.
2016 IMO Shortlist G4
Let $ABC$ be a triangle with $AB = AC \ne B$C and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.
2016 IMO Shortlist G5
Let $D$ be the foot of perpendicular from $A$ to the Euler line (the line passing through the circumcentre and the orthocentre) of an acute scalene triangle $ABC$. A circle $\omega$ with centre $S$ passes through $A$ and $D$, and it intersects sides $AB$ and $AC$ at $X$ and $Y$ respectively. Let $P$ be the foot of altitude from $A$ to $BC$, and let $M$ be the midpoint of $BC$. Prove that the circumcentre of triangle $XSY$ is equidistant from $P$ and $M$.
2016 IMO Shortlist G6
Let $ABCD$ be a convex quadrilateral with $\angle ABC = \angle ADC < 90^o$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $E$ and $F$ respectively, and meet each other at point $P$. Let $M$ be the midpoint of AC and let $ \omega$ be the circumcircle of triangle BPD. Segments BM and DM intersect $ \omega$ again at $X$ and $Y$ respectively. Denote by $Q$ the intersection point of lines $XE$ and $Y F$. Prove that $PQ \perp AC$.
2016 IMO Shortlist G7
Let $I$ be the incentre of a non-equilateral triangle $ABC$, $I_A$ be the $A$-excentre, $I'_A$ be the reflection of $I_A$ in BC, and $\ell_A$ be the reflection of line $AI'_A$ in $AI$. Define points $I_B, I'_B$ and line $\ell_B$ analogously. Let $P$ be the intersection point of $\ell_A$ and $\ell_B$.
a) Prove that $P$ lies on line $OI$ where $O$ is the circumcentre of triangle $ABC$.
b) Let one of the tangents from $P$ to the incircle of triangle $ABC$ meet the circumcircle at points $X$ and $Y$ . Show that $\angle XIY = 120^o$.
2016 IMO Shortlist G8
Let $A_1,B_1$ and $C_1$ be points on sides $BC,CA$ and $AB$ of an acute triangle $ABC$ respectively, such that $AA_1,BB_1$ and $CC_1$ are the internal angle bisectors of triangle $ABC$. Let $I$ be the incentre of triangle $ABC$, and $H$ be the orthocentre of triangle $A_1B_1C_1$. Show that $AH + BH + CH \ge AI + BI+ CI$.
2017 IMO Shortlist G1 (ITA)
Let $ABCDE$ be a convex pentagon such that $AB = BC =CD$,$ \angle EAB = \angle BCD$, and $\angle EDC =\angle CBA$. Prove that the perpendiular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.
2017 IMO Shortlist G2 (LUX) problem 4
Let $R$ and $S$ be distint points on circle $\Omega$, and let $t$ denote the tangent line to $\Omega$ at $R$. Point $R' $ is the reflection of $R$ with respect to $S$. A point $I$ is chosen on the smaller arc $RS$ of $\Omega$ so that the circumcircle $\Gamma$ of triangle $ ISR' $ intersects $t$ at two different points. Denote by $A$ the common point of $ \Gamma$ and $t$ that is closest to $R$. Line $AI$ meets $\Omega$ again at $J$. Show that $JR' $ is tangent to $ \Gamma$.
Let $O$ be the circumcenter of an acute scalene triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.
2017 IMO Shortlist G4 (DEN)
In triangle $ABC$, let $\omega$ be the excircle opposite $A$. Let $D, E$, and $F$ be the points where \omega is tangent to lines $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.
2017 IMO Shortlist G5 (UKR)
Let $ABCC_1B_1A_1$ be a convex hexagon such that $AB = BC$, and suppose that the line segments $AA_1, BB_1$, and $CC_1$ have the same perpendicular bisector. Let the diagonals $AC_1$ and $A_1C$ meet at $D$, and denote by $\omega$ the circle $ABC$. Let $\omega$ intersect the circle $A_1BC_1$ again at $E \ne B$. Prove that the lines $BB_1 $ and $DE$ intersect on $\omega$.
2017 IMO Shortlist G6 (CZE)
Let $n \ge 3$ be an integer. Two regular $n$-gons $A$ and $B$ are given in the plane. Prove that the vertices of $A$ that lie inside $B$ or on its boundary are consecutive.
(That is, prove that there exists a line separating those vertices of $A$ that lie inside $B$ or on its boundary from the other vertices of $A$.)
2017 IMO Shortlist G7 (KZA)
A convex quadrilateral $ABCD$ has an inscribed circle with center $I$. Let $I_a, I_b, I_c$, and $I_d$ be the incenters of the triangles $DAB, ABC, BCD$, and $CDA$, respectively. Suppose that the common external tangents of the circles $AI_bI_d$ and $CI_bI_d$ meet at $X$, and the common external tangents of the circles $BI_aI_c$ and $DI_aI_c $ meet at $Y$ . Prove that $\angle XIY = 90^o$.
2017 IMO Shortlist G8 (AUS)
There are $2017$ mutually external circles drawn on a blackboard, such that no two are tangent and no three share a common tangent. A tangent segment is a line segment that is a vommon tangent to two circles, starting at one tangent point and ending at the other one. Luciano is drawing tangent segments on the blackboard, one at a time, so that no tangent segment intersects any other circles or previously drawn tangent segments. Luciano keeps drawing tangent segments until no more can be drawn. Find all possible numbers of tangent segments when he stops drawing.
2018 IMO Shortlist G1 (HEL) problem 1
Let $\Gamma$ be the circumcircle of acute triangle $ABC$. Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = AE$. The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ respectively. Prove that lines $DE$ and $FG$ are either parallel or they are the same line.
2018 IMO Shortlist G2 (AUSTRALIA)
Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quarilateral $APXY$ is cyclic.
2018 IMO Shortlist G3 (SAF)
A circle $\omega$ with radius $1$ is given. A collection $T$ of triangles is called [i]good[/i], if the following conditions hold:
* each triangle from $T$ is inscribed in $\omega$;
* no two triangles from $T$ have a common interior point.
Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$.
2018 IMO Shortlist G4 (MON)
A point $T$ is chosen inside a triangle $ABC$. Let $A_1$, $B_1$, $C_1$ be the reflections of $T$ in $BC$, $CA$, and $AB$, respectively. Let $\Omega$ be the circumcircle of $\triangle A_1B_1C_1$. The lines $A_1T$, $B_1T$, $C_1T$ meet $\Omega$ again at $A_2$, $B_2$, $C_2$, respectively. Prove that lines $AA_2$, $BB_2$, $CC_2$ meet on $\Omega$.
2018 IMO Shortlist G7 (RUS)
Let $O$ be the circumcentre, and $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. Let $P$ be an arbitrary point on $\Omega$, distinct from $A$, $B$, $C$, and their antipodes in $\Omega$. Denote the circumcentres of the triangles $AOP$, $BOP$, and $COP$ by $O_A$, $O_B$, and $O_C$, respectively. The lines $\ell_A$, $\ell_B$, $\ell_C$ perpendicular to $BC$, $CA$, and $AB$ pass through $O_A$, $O_B$, and $O_C$, respectively. Prove that the circumcircle of triangle formed by $\ell_A$, $\ell_B$, and $\ell_C$ is tangent to the line $OP$.
In triangle $ABC$, point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$. Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$, respectively, such that $PQ$ is parallel to $AB$. Let $P_1$ be a point on line $PB_1$, such that $B_1$ lies strictly between $P$ and $P_1$, and $\angle PP_1C=\angle BAC$. Similarly, let $Q_1$ be the point on line $QA_1$, such that $A_1$ lies strictly between $Q$ and $Q_1$, and $\angle CQ_1Q=\angle CBA$. Prove that points $P,Q,P_1$, and $Q_1$ are concyclic.
source:
The IMO Compendium A Collection of Problems Suggested for The International Mathematical-Olympiads 1959-2009, 2nd Edition
also known as IMO SHL or IMO ISL
with aops links in the names
IMO Shortlist problems 2001 - 2018 EN in pdf with solutions
IMO Shortlist problems 1992 - 2000 EN in pdf with solutions, scanned
most of them by Orlando Döhring,
member of the IMO ShortList / LongList Project Group, in aops
IMO ISL 1968-1992
most of them by Orlando Döhring,
member of the IMO ShortList / LongList Project Group, in aops
IMO ISL 1968-1992
1993 - 2020
Let $ABC$ be an acute-angled triangle with $AB \ne AC$. The circle with diameter $BC$ intersects the sides $AB$ and $AC$ at $M$ and $N$ , respectively. Denote by $O$ the midpoint of $BC$. The bisectors of the angles $BAC$ and $MON$ intersect at $R$. Prove that the circumcircles of the triangles $BMR$ and $CNR$ have a common point lying on the line segment $BC$.
2004 IMO Shortlist G4 (POL) problem 5
2005 IMO Shortlist G4 (POL) problem 5
2005 IMO Shortlist G6 (RUS)
2005 IMO Shortlist G7 (KOR)
2006 IMO Shortlist G5 (HEL)
2006 IMO Shortlist G7 (SVK)
2006 IMO Shortlist G10 (SRB) problem 6
To each side a of a convex polygon we assign the maximum area of a triangle contained in the polygon and having a as one of its sides. Show that the sum of the areas assigned to all sides of the polygon is not less than twice the area of the polygon.
In triangle $ABC$, the angle bisector at vertex $C$ intersects the circumcircle and the perpendicular bisectors of sides $BC$ and $CA$ at points $R, P$, and $Q$, respectively. The midpoints of $BC$ and $CA$ are $S$ and $T$, respectively. Prove that triangles $RQT$ and $RPS$ have the same area.
Given an isosceles triangle $ABC$ with $AB = AC$. The midpoint of side $BC$ is denoted by $M$. Let $X$ be a variable point on the shorter arc $ MA$ of the circumcircle of triangle $ABM$. Let $T$ be the point in the angle domain $BMA$, for which $ \angle TMX = 90^o$ and $TX = BX$. Prove that $ \angle MTB - \angle CTM$ does not depend on $X$.
The diagonals of a trapezoid $ABCD$ intersect at point $P$. Point $Q$ lies between the parallel lines $BC$ and $AD$ such that $\angle AQD = \angle CQB$, and line $CD$ separates points $P$ and $Q$. Prove that $\angle BQP =\angle DAQ$.
Consider five points $A, B, C, D, E$ such that $ABCD$ is a parallelogram and BCED is a cyclic quadrilateral. Let $\ell$ be a line passing through $A$, and let $\ell $ intersect segment $ DC$ and line $BC$ at points $F$ and $G$, respectively. Suppose that $EF = EG = EC$. Prove that $\ell$ is the bisector of angle $DAB$.
Let $ABC$ be a fixed triangle, and let $A_1, B_1, C_1$ be the midpoints of sides $BC, CA, AB$, respectively. Let $P$ be a variable point on the circumcircle. Let lines $PA_1, PB_1, PC_1$ meet the circumcircle again at $A', B', C' $ respectively. Assume that the points $A, B, C, A', B', C' $ are distinct, and lines $AA', BB', CC' $ form a triangle. Prove that the area of this triangle does not depend on $P$.
Determine the smallest positive real number k with the following property. Let $ABCD$ be a convex quadrilateral, and let points $A_1, B_1, C_1$ and $D_1$ lie on sides $AB, BC,CD$ and $DA$, respectively. Consider the areas of triangles $AA_1D_1, BB_1A_1, CC_1B_1$, and $DD_1C_1$; let $S $ be the sum of the two smallest ones, and let $S_1$ be the area of quadrilateral $A_1B_1C_1D_1$. Then we always have $ kS_1 \ge S $.
Given an acute triangle $ABC$ with angles $ \alpha, \beta $ and $ \gamma $ at vertices $A, B$ and $C$, respectively, such that $ \beta > \gamma $ . Point $I $ is the incenter, and $R$ is the circumradius. Point $D $ is the foot of the altitude from vertex $A$. Point $K$ lies on line $AD$ such that $AK = 2R$, and $D$ separates $A$ and $K$. Finally, lines $DI$ and $KI$ meet sides $AC$ and $BC$ at $E$ and $F$, respectively. Prove that if $IE = IF$ then $\beta \le 3 \gamma $.
Point $P$ lies on side $AB $ of a convex quadrilateral $ABCD$. Let $\omega$ be the incircle of triangle $CPD$, and let $I$ be its incenter. Suppose that $\omega$ is tangent to the incircles of triangles $APD$ and $BPC$ at points $K$ and $L$, respectively. Let lines $AC$ and $BD$ meet at $E$, and let lines $AK$ and $BL$ meet at $F$. Prove that points $E, I$, and $F$ are collinear.
In an acute-angled triangle $ABC$, point $H$ is the orthocentre and $A_o, B_o, C_o$ are the midpoints of the sides $BC, CA, AB$, respectively. Consider three circles passing through $H$: $\omega_a$ around $A_o, \omega_b$ around $B_o$ and $\omega_c$ around $C_o$. The circle $\omega_a$ intersects the line $BC$ at $A_1$ and $A_2$; $\omega_b$ intersects$ CA$ at $B_1$ and $B_2$; $\omega_c$ intersects $AB$ at $C_1$ and $C_2$. Show that the points $A_1, A_2, B_1, B_2, C_1, C_2$ lie on a circle.
Given trapezoid $ABCD$ with parallel sides $AB$ and $CD$, assume that there exist points $ E$ on line $ BC$ outside segment $BC$, and F inside segment $AD$, such that $\angle DAE = \angle CBF$. Denote by $I$ the point of intersection of $CD$ and $EF$, and by $J$ the point of intersection of $AB$ and $EF$. Let $K$ be the midpoint of segment $EF$; assume it does not lie on line $AB$. Prove that $I$ belongs to the circumcircle of $ABK$ if and only if $K$ belongs to the circumcircle of $CDJ$.
Let $ABCD$ be a convex quadrilateral and let $P$ and $ Q$ be points in $ABCD$ such that $ PQDA$ and $QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $E$ on the line segment $ PQ$ such that $ \angle PAE = \angle QDE$ and $\angle PBE =\angle QCE$. Show that the quadrilateral $ABCD$ is cyclic.
In an acute triangle $ABC$ segments$ BE $ and $CF $ are altitudes. Two circles passing through the points $A$ and $F$ are tangent to the line $BC$ at the points $P$ and $Q$ so that $B$ lies between $C$ and $Q$. Prove that the lines $PE$ and $QF$ intersect on the circumcircle of triangle $AEF$.
2008 IMO Shortlist G5 (NLD)
Let$ k$ and $ n $be integers with $ 0 \le k \le n - 2$ . Consider a set $L $ of $n $ lines in the plane such that no two of them are parallel and no three have a common point. Denote by $I $ the set of intersection points of lines in $L$. Let $O$ be a point in the plane not lying on any line of $L$. A point $X \in I $ is colored red if the open line segment $ OX $ intersects at most $k$ lines in $L$. Prove that $ I$ contains at least $ \frac{1}{2} (k+ 1)(k + 2) $red points.
2008 IMO Shortlist G6 (SRB)
There is given a convex quadrilateral $ABCD$. Prove that there exists a point $P$ inside the quadrilateral such that $ \angle PAB + \angle PDC = \angle PBC + \angle PAD = \angle PCD + \angle PBA = \angle PDA + \angle PCB = 90^o$ if and only if the diagonals $AC$ and $BD $ are perpendicular.
Let $ABCD$ be a convex quadrilateral with $AB \ne BC$. Denote by $\omega_1$ and $\omega_2$ the incircles of triangles $ABC$ and $ADC$. Suppose that there exists a circle $ \omega$ inscribed in angle $ABC$, tangent to the extensions of line segments $AD$ and $CD$. Prove that the common external tangents of $\omega_1 $ and $\omega_2 $ intersect on $ \omega$.
Let ABC be a triangle with $AB = AC$. The angle bisectors of $A$ and $B$ meet the sides $BC$ and $AC$ in $D$ and $E$, respectively. Let $K$ be the incenter of triangle $ADC$. Suppose that $\angle BEK = 45^o$. Find all possible values of $\angle BAC$.
Let $ABC$ be a triangle with circumcenter $O$. The points $P$ and $Q$ are interior points of the sides $CA$ and $AB$, respectively. The circle $k $ passes through the midpoints of the segments $BP, CQ$, and $PQ$. Prove that if the line $PQ $ is tangent to circle $ k$ then $ OP = OQ$.
Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$ , respectively. Let $G $ be the point where the lines $BY $ and $CZ $ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCY R$ and $BCSZ$ are parallelograms. Prove that $GR = GS$.
2009 IMO Shortlist G4 (UNK)
Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and$ BD$ meet at $E $ and the lines $AD $ and $ BC $ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E, G,$ and $ H$.
Let $P$ be a polygon that is convex and symmetric to some point $O$. Prove that for some parallelogram $R$ satisfying $P \subset R$ we have $\frac{|R|}{|P| }\le \sqrt{2}$ where $ |R|$ and $ |P| $ denote the area of the sets $R$ and $P$, respectively.
2009 IMO Shortlist G6 (UKR)
Let the sides $AD$ and $BC$ of the quadrilateral $ABCD$ (such that $AB $ is not parallel to $CD$) intersect at point $P$. Points $O_1 $ and $O_2 $ are the circumcenters and points $H_1 $ and $H_2$ are the orthocenters of triangles $ABP$ and $DCP$, respectively. Denote the midpoints of segments $O_1H_1$ and $O_2H_2$ by $E_1$ and $E_2$, respectively. Prove that the perpendicular from $E_1 $ on $ CD$, the perpendicular from $E_2$ on $AB$ and the line $H_1H_2$ are concurrent.
Let $ABC $ be a triangle with incenter $ I$ and let $X, Y $ and $ Z$ be the incenters of the triangles $BIC, CIA$ and $AIB$, respectively. Let the triangle $ XY Z $ be equilateral. Prove that $ABC$ is equilateral too.
2009 IMO Shortlist G8 (BGR)
Let $ABCD$ be a circumscribed quadrilateral. Let $g$ be a line through $A$ which meets the segment $BC$ in $M$ and the line $CD $ in $N$. Denote by $ I_1, I_2$, and $ I_3$ the incenters of $\vartriangle ABM, \vartriangle MNC$, and $ \vartriangle NDA$, respectively. Show that the orthocenter of $\vartriangle I_1I_2I_3$ lies on $g$.
2010 IMO Shortlist G1 (UNK)
Let $ABC $ be an acute triangle with $D,E, F$ the feet of the altitudes lying on $BC,CA,AB$ respectively. One of the intersection points of the line $EF$ and the circumcircle is $P$. The lines $BP$ and $DF$ meet at point $Q$. Prove that $AP = AQ$.
Point $P$ lies inside triangle $ABC$. Lines $AP, BP, CP$ meet the circumcircle of $ABC$ again at points $K, L, M$, respectively. The tangent to the circumcircle at $C$ meets line $AB$ at $S$. Prove that $SC = SP$ if and only if $MK = ML$.
Let $A_1A_2...A_n $ be a convex polygon. Point $P$ inside this polygon is chosen so that its projections $P_1, ..., P_n$ onto lines $A_1A_2, ... , A_nA_1$ respectively lie on the sides of the polygon. Prove that for arbitrary points $X_1, ... , X_n$ on sides $A_1A_2, ... , A_nA_1$ respectively, $ max \{ \frac{X_1X_2}{P_1P_2},... , \frac{X_nX_1}{P_nP_1}\} \ge 1$.
Let $I$ be the incenter of a triangle $ABC$ and $ \Gamma $ be its circumcircle. Let the line $AI$ intersect $ \Gamma$ at a point $D \ne A$. Let $F$ and $E$ be points on side $BC$ and arc $BDC$ respectively such that $\angle BAF = \angle CAE < \frac{1}{2} \angle BAC$. Finally, let $G $ be the midpoint of the segment $IF$. Prove that the lines $DG$ and $EI$ intersect on $\Gamma$.
Let $ABCDE $ be a convex pentagon such that $BC// AE, AB = BC + AE$, and $ \angle ABC = \angle CDE$. Let $M$ be the midpoint of $CE$, and let $O $ be the circumcenter of triangle $BCD$. Given that $\angle DMO =90^o$ , prove that $2 \angle BDA = \angle CDE$.
The vertices $X, Y , Z$ of an equilateral triangle $XYZ$ lie respectively on the sides $BC, CA, AB$ of an acute-angled triangle $ABC$. Prove that the incenter of triangle $ABC$ lies inside triangle $XYZ$.
2010 IMO Shortlist G7 (HUN)
Three circular arcs $\gamma_1, \gamma_2,$ and $\gamma_3$ connect the points $A$ and $C.$ These arcs lie in the same half-plane defined by line $AC$ in such a way that arc $\gamma_2$ lies between the arcs $\gamma_1$ and $\gamma_3.$ Point $B$ lies on the segment $AC.$ Let $h_1, h_2$, and $h_3$ be three rays starting at $B,$ lying in the same half-plane, $h_2$ being between $h_1$ and $h_3.$ For $i, j = 1, 2, 3,$ denote by $V_{ij}$ the point of intersection of $h_i$ and $\gamma_j$ (see the Figure below). Denote by $\widehat{V_{ij}V_{kj}}\widehat{V_{kl}V_{il}}$ the curved quadrilateral, whose sides are the segments $V_{ij}V_{il},$ $V_{kj}V_{kl}$ and arcs $V_{ij}V_{kj}$ and $V_{il}V_{kl}.$ We say that this quadrilateral is $circumscribed$ if there exists a circle touching these two segments and two arcs. Prove that if the curved quadrilaterals $\widehat{V_{11}V_{21}}\widehat{V_{22}V_{12}}, \widehat{V_{12}V_{22}}\widehat{V_{23}V_{13}},\widehat{V_{21}V_{31}}\widehat{V_{32}V_{22}}$ are circumscribed, then the curved quadrilateral $\widehat{V_{22}V_{32}}\widehat{V_{33}V_{23}}$ is circumscribed, too.
2011 IMO Shortlist G1 (EST)
Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose center $L$ lies on the side $BC$. Suppose that $ \omega$ is tangent to $AB$ at $B'$ and to $AC$ at $C'$ . Suppose also that the circumcenter $O$ of the triangle $ABC$ lies on the shorter arc $B' C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points.
Let $A_1A_2A_3A_4 $ be a non-cyclic quadrilateral. Let $O_1 $ and $r_1 $ be the circumcenter and the circumradius of the triangle $A_2A_3A_4$. Define $O_2, O_3, O_4$ and $r_2, r_3, r_4$ in a similar way. Prove that $ \frac{1}{O_1A_1^2 - r_1^2}+ \frac{1}{O_2A_2^2 - r_2^2}+\frac{1}{O_3A_3^2 - r_3^2}+\frac{1}{O_4A_4^2 - r_4^2}= 0$.
Let A$BCD$ be a convex quadrilateral whose sides $AD$ and $BC$ are not parallel. Suppose that the circles with diameters $AB$ and $CD$ meet at points $E$ and $F$ inside the quadrilateral. Let $\omega_E$ be the circle through the feet of the perpendiculars from $E$ to the lines $AB, BC$, and $CD$. Let $ \omega_F$ be the circle through the feet of the perpendiculars from $F$ to the lines $CD, DA$, and $AB$. Prove that the midpoint of the segment $EF$ lies on the line through the two intersection points of $ \omega_E$ and $\omega_F$ .
Let $ABC$ be an acute triangle with circumcircle $\Omega$. Let $B_0$ be the midpoint of $AC$ and let $C_0$ be the midpoint of $AB$. Let $D$ be the foot of the altitude from $A$, and let $G$ be the centroid of the triangle $ABC$. Let $\omega $ be a circle through $B_0$ and $C_0$ that is tangent to the circle $\Omega$ at a point $X \ne A$. Prove that the points $D, G$, and $X$ are collinear.
Let $ABC$ be a triangle with $AB = AC$, and let $D$ be the midpoint of $AC$. The angle bisector of $ \angle BAC$ intersects the circle through $D, B$, and $C$ in a point $E$ inside the triangle $ABC$. The line BD intersects the circle through $A, E$, and $B$ in two points $B$ and $F$. The lines $AF$ and $BE$ meet at a point $I$, and the lines $CI$ and $BD$ meet at a point $K$. Show that $I$ is the incenter of triangle $KAB$.
Let $ABCDEF$ be a convex hexagon all of whose sides are tangent to a circle $\omega$ with center O. Suppose that the circumcircle of triangle $ACE$ is concentric with $\omega$ . Let $J$ be the foot of the perpendicular from $B$ to $CD$. Suppose that the perpendicular from $B$ to $DF$ intersects the line $EO$ at a point $K$. Let $L$ be the foot of the perpendicular from $K$ to $DE$. Prove that $DJ = DL$.
2011 IMO Shortlist G8 (JPN) problem 6
Let $ABC$ be an acute triangle with circumcircle $\omega$. Let $t$ be a tangent line to $ \omega.$ Let $t_a, t_b$, and $t_c$ be the lines obtained by reflecting $t$ in the lines $BC, CA$, and $AB$, respectively. Show that the circumcircle of the triangle determined by the lines $t_a, t_b$, and $t_c$ is tangent to the circle $\omega$.
2012 IMO Shortlist G1 (HEL) problem 1
In the triangle $ABC$ the point $J$ is the center of the excircle opposite to $A$. This excircle is tangent to the side $BC$ at $M$, and to the lines $AB$ and $AC$ at $K$ and $L$ respectively. The lines $LM$ and $BJ$ meet at $F$, and the lines $KM$ and $CJ$ meet at $G$. Let $S$ be the point of intersection of the lines $AF$ and $BC$, and let $T$ be the point of intersection of the lines $AG$ and $BC$. Prove that $M$ is the midpoint of $ST$.
Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ meet at $E$. The extensions of the sides $AD$ and $BC$ beyond $A$ and $B$ meet at $F$. Let $G$ be the point such that $ECGD$ is a parallelogram, and let $H$ be the image of $E$ under reflection in $AD$. Prove that $D, H, F, G$ are concyclic.
2012 IMO Shortlist G3
In an acute triangle $ABC$ the points $D, E$ and $F$ are the feet of the altitudes through $A, B$ and $C$ respectively. The incenters of the triangles $AEF$ and $BDF$ are $I_1$ and $I_2$ respectively; the circumcenters of the triangles $ACI_1$ and $BCI_2$ are $O_1 $ and $O_2$ respectively. Prove that $I_1I_2$ and $O_1O_2$ are parallel.
2012 IMO Shortlist G4
Let $ABC$ be a triangle with $AB \ne AC$ and circumcenter $O$. The bisector of $\angle BAC$ intersects $BC$ at $D$. Let $E$ be the reflection of $D$ with respect to the midpoint of $BC$. The lines through $D$ and $E$ perpendicular to $BC$ intersect the lines $AO$ and $AD$ at $X$ and $Y$ respectively. Prove that the quadrilateral $BXCY$ is cyclic.
2012 IMO Shortlist G5 (CZE) problem 5
Let $ABC$ be a triangle with $\angle BCA = 90^o$, and let $C_0$ be the foot of the altitude from $C$. Choose a point $X$ in the interior of the segment $CC_0$, and let $K, L$ be the points on the segments $AX,BX$ for which $BK = BC$ and $AL = AC$ respectively. Denote by $M$ the intersection of $AL$ and $BK$. Show that $MK = ML$.
Let $ABC$ be a triangle with circumcenter $O$ and incenter $I$. The points $D, E$ and $F$ on the sides $BC, CA$ and $AB$ respectively are such that $BD+ BF=CA$ and $CD + CE = AB$. The circumcircles of the triangles $BFD$ and $CDE$ intersect at $P \ne D$. Prove that $OP = OI$.
2012 IMO Shortlist G7
Let $ABCD$ be a convex quadrilateral with non-parallel sides $BC$ and $AD$. Assume that there is a point $E$ on the side $BC$ such that the quadrilaterals $ABED$ and $AECD$ are circumscribed. Prove that there is a point $F$ on the side $AD$ such that the quadrilaterals $ABCF$ and $BCDF$ are circumscribed if and only if $AB$ is parallel to $CD$.
2012 IMO Shortlist G8 (ROU)
Let $ABC$ be a triangle with circumcircle $\omega$ and $\ell$ a line without common points with $\omega$. Denote by $P$ the foot of the perpendicular from the center of $\omega$ to $\ell$. The side-lines $BC,CA,AB$ intersect $\ell$ at the points $X, Y, Z$ different from $P$. Prove that the circumcircles of the triangles $AXP,BYP$ and $CZP$ have a common point different from $P$ or are mutually tangent at $P$.
Let $ABC$ be an acute-angled triangle with orthocenter $H$, and let $W$ be a point on side $BC$. Denote by $M$ and $N$ the feet of the altitudes from $B$ and $C$, respectively. Denote by $\omega_1$ the circumcircle of $BWN$, and let $X$ be the point on $\omega_1$ which is diametrically opposite to $W$. Analogously, denote by $\omega_2$ the circumcircle of $CWM$, and let $Y$ be the point on $\omega_2$ which is diametrically opposite to $W$. Prove that $X, Y$ and $H$ are collinear.
Let $\omega$ be the circumcircle of a triangle $ABC$. Denote by $M$ and $N$ the midpoints of the sides $AB$ and $AC$, respectively, and denote by $T$ the midpoint of the arc $BC$ of $\omega$ not containing $A$. The circumcircles of the triangles $AMT$ and $ANT$ intersect the perpendicular bisectors of $AC$ and $AB$ at points $X$ and $Y$ , respectively; assume that $X$ and $Y$ lie inside the triangle $ABC$. The lines $MN$ and $XY$ intersect at $K$. Prove that $KA= KT$.
2013 IMO Shortlist G3 (SRB)
In a triangle $ABC$, let $D$ and $E$ be the feet of the angle bisectors of angles $A$ and $B$, respectively. A rhombus is inscribed into the quadrilateral $AEDB$ (all vertices of the rhombus lie on different sides of $AEDB$). Let $\phi$ be the non-obtuse angle of the rhombus. Prove that $\phi \le max\{\angle BAC,\angle ABC\}$.
2013 IMO Shortlist G4 (GEO)
Let $ABC$ be a triangle with $\angle B > \angle C$. Let $P$ and $Q$ be two different points on line $AC$ such that $\angle PBA =\angle QBA =\angle ACB$ and $A$ is located between $P$ and $C$. Suppose that there exists an interior point $D$ of segment $BQ$ for which $PD = PB$. Let the ray $AD$ intersect the circle $ABC$ at $R \ne A$. Prove that $QB = QR$.
2013 IMO Shortlist G5 (UKR)
Let $ABCDEF$ be a convex hexagon with $AB = DE, BC = EF, CD =FA$, and $\angle A - \angle D =\angle C - \angle F =\angle E- \angle B$. Prove that the diagonals $AD,BE$, and $CF$ are concurrent.
2013 IMO Shortlist G6 (RUS) problem 3
Let the excircle of the triangle $ABC$ lying opposite to $A$ touch its side $BC$ at the point $A_1$. Define the points $B_1$ and $C_1$ analogously. Suppose that the circumcentre of the triangle $A_1B_1C_1$ lies on the circumcircle of the triangle $ABC$. Prove that the triangle $ABC$ is right-angled.
The points $P$ and $Q$ are chosen on the side $BC$ of an acute-angled triangle $ABC$ so that $\angle PAB =\angle ACB$ and $\angle QAC =\angle CBA$. The points $M$ and $N$ are taken on the rays $AP$ and $AQ$, respectively, so that $AP = PM$ and $AQ =QN$. Prove that the lines $BM$ and $CN$ intersect on the circumcircle of the triangle $ABC$.
Let $ABC$ be a triangle. The points $K, L$, and $M$ lie on the segments $BC, CA$, and $AB$, respectively, such that the lines $AK, BL$, and $CM$ intersect in a common point. Prove that it is possible to choose two of the triangles $ALM, BMK$, and $CKL$ whose inradii sum up to at least the inradius of the triangle $ABC$.
2014 IMO Shortlist G3 (RUS)
Let $\Omega$ and $O$ be the circumcircle and the circumcentre of an acute-angled triangle $ABC$ with $AB > BC$. The angle bisector of $\angle ABC$ intersects $\Omega$ at $M \ne B$. Let $\Gamma$ be the circle with diameter $BM$. The angle bisectors of $\angle AOB$ and $\angle BOC$ intersect $\Gamma$ at points $P$ and $Q$, respectively. The point $R$ is chosen on the line $PQ$ so that $BR= MR$. Prove that $BR // AC$. (Here we always assume that an angle bisector is a ray.)
2014 IMO Shortlist G4 (UNK)
Consider a fixed circle $\Gamma$ with three fixed points $A, B$, and $C$ on it. Also, let us fix a real number $\lambda \in (0, 1)$. For a variable point $P \notin \{A,B,C\}$ on $\Gamma$, let $M$ be the point on the segment $CP $ such that $CM= \lambda \cdot CP$. Let $Q$ be the second point of intersection of the circumcircles of the triangles $AMP$ and $BMC$. Prove that as $P$ varies, the point $Q$ lies on a fixed circle.
2014 IMO Shortlist G5 (IRN) problem 3
Let $ABCD$ be a convex quadrilateral with $\angle B = \angle D =90^o$. Point H is the foot of the perpendicular from $A$ to $BD$. The points $S$ and $T$ are chosen on the sides $AB$ and $AD$, respectively, in such a way that $H$ lies inside triangle $SCT$ and $ \angle SHC - \angle BSC= 90^o , \angle THC - \angle DTC = 90^o$ . Prove that the circumcircle of triangle $SHT$ is tangent to the line $BD$.
2014 IMO Shortlist G6 (IRN)
Let $ABC$ be a fixed acute-angled triangle. Consider some points $E$ and $F$ lying on the sides $AC$ and $AB$, respectively, and let $M$ be the midpoint of $EF$. Let the perpendicular bisector of $EF$ intersect the line $BC$ at $K$, and let the perpendicular bisector of $MK$ intersect the lines $AC$ and $AB$ at $S$ and $T$, respectively. We call the pair $(E, F)$ interesting, if the quadrilateral $KSAT$ is cyclic. Suppose that the pairs $(E_1, F_1)$ and $(E_2, F_2)$ are interesting. Prove that $\frac{E_1E_2}{AB}= \frac{F_1F_2}{AC}$.
2014 IMO Shortlist G7 (USA)
Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I.$ Let the line passing through $I$ and perpendicular to $CI$ intersect the segment $BC$ and the arc $BC$ (not containing $A$) of $\Omega$ at points $U$ and $V$ , respectively. Let the line passing through $U$ and parallel to $AI $ intersect $AV$ at $X$, and let the line passing through $V$ and parallel to $AI$ intersect $AB$ at $Y$ . Let $W$ and $Z$ be the midpoints of $AX$ and $BC$, respectively. Prove that if the points $I, X$, and $Y$ are collinear, then the points $I, W$, and $Z$ are also collinear.
1993 IMO Shortlist ISL2 (CAN 2)
Let triangle $ABC$ be such that its circumradius $R$ is equal to $1$. Let $r$ be the inradius of $ABC$ and let $p$ be the inradius of the orthic triangle $A'B'C'$ of triangle $ABC$. Prove that $p \le 1- \frac{1}{3} (1+r)^2$.
Remark. The orthic triangle is the triangle whose vertices are the feet of the altitudes of ABC.
1993 IMO Shortlist ISL3 (ESP 1)
Consider the triangle $ABC$, its circumcircle $k$ with center $O$ and radius $R$, and its incircle with center $I$ and radius $r$. Another circle $k_c$ is tangent to the sides $CA,CB$ at $D,E$, respectively, and it is internally tangent to $k$. Show that the incenter $I$ is the midpoint of $DE$.
In the triangle $ABC$, let $D,E$ be points on the side $BC$ such that $\angle BAD=\angle CAE$. If $M,N$ are, respectively, the points of tangency with $BC$ of the incircles of the triangles ABD and ACE, show that $\frac{1}{MB}+\frac{1}{MD}=\frac{1}{NC}+\frac{1}{NE}$.
The vertices $D,E,F$ of an equilateral triangle lie on the sides $BC,CA,AB$ respectively of a triangle $ABC$. If $a,b,c$ are the respective lengths of these sides, and $S$ the area of $ABC$, prove that $DE \ge \frac{2\sqrt2 S}{\sqrt{a%2+b%2+c^2+4\sqrt3S}}$.
1993 IMO Shortlist ISL15 (MKD 1) problem 4
For three points $A,B,C$ in the plane we define $m(ABC)$ to be the smallest length of the three altitudes of the triangle $ABC$, where in the case of $A,B,C$ collinear, $m(ABC)=0$. Let $A,B,C$ be given points in the plane. Prove that for any point $X$ in the plane, $m(ABC) \le m(ABX)+m(AXC)+m(XBC)$.
A circle $S$ is said to cut a circle $S$ diametrally if their common chord is a diameter of $S$ . Let $SA,SB,SC $ be three circles with distinct centers $A,B,C$ respectively. Prove that $A,B,C$ are collinear if and only if there is no unique circle S that cuts each of $SA,SB,SC$ diametrally. Prove further that if there exists more than one circle S that cuts each of $SA,SB,SC$ diametrally, then all such circles pass through two fixed points. Locate these points in relation to the circles $SA,SB,SC$.
$A,B,C,D$ are four points in the plane, with $C,D$ on the same side of the line $AB$, such that $AC\cdot BD = AD\cdot BC$ and $\angle ADB = 90^o +\angle ACB$. Find the ratio $\frac{AB\cdot CD}{ AC\cdot BD }$, and prove that circles $ACD,BCD$ are orthogonal.
(Intersecting circles are said to be orthogonal if at either common point their tangents are perpendicular.)
1994 IMO Shortlist G1 (FRA)
A semicircle $\Gamma$ is drawn on one side of a straight line $\ell$. $C$ and D are points on $\Gamma$ . The tangents to $\Gamma$ at C and D meet $\ell$ at $B$ and $A$ respectively, with the center of the semicircle between them. Let $E$ be the point of intersection of $AC$ and $BD$, and $F$ the point on $\ell$ such that $EF$ is perpendicular to $\ell$. Prove that $EF$ bisects $\angle CFD$.
1994 IMO Shortlist G2 (UKR)
$ABCD$ is a quadrilateral with BC parallel to $AD$. $M$ is the midpoint of $CD$, $P$ that of $MA$ and $Q$ that of $MB$. The lines $DP$ and $CQ$ meet at $N$. Prove that $N$ is not outside $ABCD$.
by Vyacheslav Yasinskiy
1994 IMO Shortlist G3 (RUS)
A circle $\omega$ is tangent to two parallel lines $\ell_1$ and $\ell_2$. A second circle $\omega_1$ is tangent to $\ell_1$ at $A$ and to $\omega$ externally at $C$. A third circle $\omega_2$ is tangent to $\ell_2$ at $B$, to $\omega$ externally at $D$, and to $\omega_1$ externally at $E$. $AD$ intersects $BC$ at $Q$. Prove that $Q$ is the circumcenter of triangle $CDE$.
1994 IMO Shortlist G4 (AUS-ARM) problem 2
$N$ is an arbitrary point on the bisector of $\angle BAC$. $P$ and $O$ are points on the lines $AB$ and $AN$, respectively, such that $\angle ANP=90^o =\angle APO$. $Q$ is an arbitrary point on $NP$, and an arbitrary line through $Q$ meets the lines $AB$ and $AC$ at $E$ and $F$ respectively. Prove that $\angle OQE= 90^o$ if and only if $QE=QF$.
A line $\ell$ does not meet a circle $\omega$ with center $O$. $E$ is the point on $\ell$ such that $OE$ is perpendicular to $\ell$. $M$ is any point on $\ell$ other than $E$. The tangents from $M$ to $\omega$ touch it at $A$ and $B$. $C$ is the point on $MA$ such that $EC$ is perpendicular to $MA$. $D$ is the point on $MB$ such that $ED$ is perpendicular to $MB$. The line $CD$ cuts $OE$ at $F$. Prove that the location of $F$ is independent of that of $M$.
1995 IMO Shortlist G1 (BGR) problem 1
Let $A,B,C$, and $D$ be distinct points on a line, in that order. The circles with diameters $AC$ and $BD$ intersect at $X$ and $Y$. $O$ is an arbitrary point on the line $XY$ but not on $AD$. $CO$ intersects the circle with diameter $AC$ again at $M$, and $BO$ intersects the other circle again at $N$. Prove that the lines $AM,DN$, and $XY$ are concurrent.
Let $A,B$, and $C$ be noncollinear points. Prove that there is a unique point $X$ in the plane of $ABC$ such that $XA^2+XB^2+AB^2 = XB^2+XC^2+BC^2 = XC^2+XA^2+CA^2$.
The incircle of $ABC$ touches $BC, CA$, and $AB$ at $D, E$, and $F$ respectively. $X$ is a point inside $ABC$ such that the incircle of $XBC$ touches $BC$ at $D$ also, and touches $CX$ and $XB$ at $Y$ and $Z$, respectively. Prove that $EFZY$ is a cyclic quadrilateral.
1995 IMO Shortlist G4 (UKR)
An acute triangle $ABC$ is given. Points $A_1$ and $A_2$ are taken on the side $BC$ (with $A_2$ between $A_1$ and $C$), $B_1$ and $B_2$ on the side AC (with B_2 between $B_1$ and $A$), and $C_1$ and $C_2$ on the side $AB$ (with $C_2$ between $C_1$ and $B$) such that $\angle AA_1A_2 = \angle AA_2A_1 = \angle BB_1B_2 = \angle BB_2B_1 = \angle CC_1C_2 = \angle CC_2C_1$. The lines $AA_1,BB_1$, and $CC_1$ form a triangle, and the lines $AA_2,BB_2$, and $CC_2$ form a second triangle. Prove that all six vertices of these two triangles lie on a single circle.
Let $ABCDEF$ be a convex hexagon with $AB = BC =CD, DE = EF = FA$, and $ \angle BCD = \angle EFA = \pi /3$ (that is, $60^o$). Let $G$ and $H$ be two points interior to the hexagon such that angles $AGB$ and $DHE$ are both $2\pi /3$ (that is,$120^o$). Prove that $AG+GB+GH+DH +HE \ge CF$.
Let $A_1A_2A_3A_4$ be a tetrahedron, $G$ its centroid, and $A_1',A_2',A_3'$, and $A_4'$ the points where the circumsphere of $A_1A_2A_3A_4$ intersects $GA_1,GA_2,GA_3$, and $GA_4$, respectively. Prove that $GA_1 \cdot GA_2 \cdot GA_3 \cdot GA_4 \le GA_1' \cdot GA_2' \cdot GA_3' \cdot GA_4'$ and $\frac{1}{GA_1'}+\frac{1}{GA_2'}+\frac{1}{GA_3'} +\frac{1}{GA_4' }\le \frac{1}{GA_1}+\frac{1}{GA_2}+\frac{1}{GA_3}+\frac{1}{GA_4}$.
$O$ is a point inside a convex quadrilateral $ABCD$ of area $S$. $K, L, M$, and $N$ are interior points of the sides $AB, BC, CD$, and $DA$ respectively. If $OKBL$ and $OMDN$ are parallelograms, prove that $\sqrt{S} \ge \sqrt{S_1} +\sqrt{S_2}$, where $S_1$ and $S_2$ are the areas of $ONAK$ and $OLCM$ respectively.
1995 IMO Shortlist G8 (COL)
Let $ABC$ be a triangle. A circle passing through $B$ and $C$ intersects the sides $AB$ and $AC$ again at $C'$ and $B'$, respectively. Prove that $BB',CC'$, and $HH'$ are concurrent, where $H$ and $ H'$ are the orthocenters of triangles $ABC$ and $AB'C'$ respectively.
2001 IMO
Shortlist G1 (UKR)
Remark. The orthic triangle is the triangle whose vertices are the feet of the altitudes of ABC.
1993 IMO Shortlist ISL3 (ESP 1)
Consider the triangle $ABC$, its circumcircle $k$ with center $O$ and radius $R$, and its incircle with center $I$ and radius $r$. Another circle $k_c$ is tangent to the sides $CA,CB$ at $D,E$, respectively, and it is internally tangent to $k$. Show that the incenter $I$ is the midpoint of $DE$.
by Francisco Bellot Rosado
1993 IMO Shortlist ISL4 (ESP 2)In the triangle $ABC$, let $D,E$ be points on the side $BC$ such that $\angle BAD=\angle CAE$. If $M,N$ are, respectively, the points of tangency with $BC$ of the incircles of the triangles ABD and ACE, show that $\frac{1}{MB}+\frac{1}{MD}=\frac{1}{NC}+\frac{1}{NE}$.
by Francisco Bellot Rosado
1993 IMO Shortlist ISL14 (ISR 1)The vertices $D,E,F$ of an equilateral triangle lie on the sides $BC,CA,AB$ respectively of a triangle $ABC$. If $a,b,c$ are the respective lengths of these sides, and $S$ the area of $ABC$, prove that $DE \ge \frac{2\sqrt2 S}{\sqrt{a%2+b%2+c^2+4\sqrt3S}}$.
1993 IMO Shortlist ISL15 (MKD 1) problem 4
For three points $A,B,C$ in the plane we define $m(ABC)$ to be the smallest length of the three altitudes of the triangle $ABC$, where in the case of $A,B,C$ collinear, $m(ABC)=0$. Let $A,B,C$ be given points in the plane. Prove that for any point $X$ in the plane, $m(ABC) \le m(ABX)+m(AXC)+m(XBC)$.
by D. Dimovski
1993 IMO Shortlist ISL21 (UNK 1)A circle $S$ is said to cut a circle $S$ diametrally if their common chord is a diameter of $S$ . Let $SA,SB,SC $ be three circles with distinct centers $A,B,C$ respectively. Prove that $A,B,C$ are collinear if and only if there is no unique circle S that cuts each of $SA,SB,SC$ diametrally. Prove further that if there exists more than one circle S that cuts each of $SA,SB,SC$ diametrally, then all such circles pass through two fixed points. Locate these points in relation to the circles $SA,SB,SC$.
by Chirstopher Bradley
1993 IMO Shortlist ISL22 (UNK 2) problem 2$A,B,C,D$ are four points in the plane, with $C,D$ on the same side of the line $AB$, such that $AC\cdot BD = AD\cdot BC$ and $\angle ADB = 90^o +\angle ACB$. Find the ratio $\frac{AB\cdot CD}{ AC\cdot BD }$, and prove that circles $ACD,BCD$ are orthogonal.
(Intersecting circles are said to be orthogonal if at either common point their tangents are perpendicular.)
by David Monk
1994 IMO Shortlist G1 (FRA)
A semicircle $\Gamma$ is drawn on one side of a straight line $\ell$. $C$ and D are points on $\Gamma$ . The tangents to $\Gamma$ at C and D meet $\ell$ at $B$ and $A$ respectively, with the center of the semicircle between them. Let $E$ be the point of intersection of $AC$ and $BD$, and $F$ the point on $\ell$ such that $EF$ is perpendicular to $\ell$. Prove that $EF$ bisects $\angle CFD$.
1994 IMO Shortlist G2 (UKR)
$ABCD$ is a quadrilateral with BC parallel to $AD$. $M$ is the midpoint of $CD$, $P$ that of $MA$ and $Q$ that of $MB$. The lines $DP$ and $CQ$ meet at $N$. Prove that $N$ is not outside $ABCD$.
by Vyacheslav Yasinskiy
A circle $\omega$ is tangent to two parallel lines $\ell_1$ and $\ell_2$. A second circle $\omega_1$ is tangent to $\ell_1$ at $A$ and to $\omega$ externally at $C$. A third circle $\omega_2$ is tangent to $\ell_2$ at $B$, to $\omega$ externally at $D$, and to $\omega_1$ externally at $E$. $AD$ intersects $BC$ at $Q$. Prove that $Q$ is the circumcenter of triangle $CDE$.
1994 IMO Shortlist G4 (AUS-ARM) problem 2
$N$ is an arbitrary point on the bisector of $\angle BAC$. $P$ and $O$ are points on the lines $AB$ and $AN$, respectively, such that $\angle ANP=90^o =\angle APO$. $Q$ is an arbitrary point on $NP$, and an arbitrary line through $Q$ meets the lines $AB$ and $AC$ at $E$ and $F$ respectively. Prove that $\angle OQE= 90^o$ if and only if $QE=QF$.
by H. Lausch & G. Tonoyan
1994 IMO Shortlist G5 (CYP)A line $\ell$ does not meet a circle $\omega$ with center $O$. $E$ is the point on $\ell$ such that $OE$ is perpendicular to $\ell$. $M$ is any point on $\ell$ other than $E$. The tangents from $M$ to $\omega$ touch it at $A$ and $B$. $C$ is the point on $MA$ such that $EC$ is perpendicular to $MA$. $D$ is the point on $MB$ such that $ED$ is perpendicular to $MB$. The line $CD$ cuts $OE$ at $F$. Prove that the location of $F$ is independent of that of $M$.
1995 IMO Shortlist G1 (BGR) problem 1
Let $A,B,C$, and $D$ be distinct points on a line, in that order. The circles with diameters $AC$ and $BD$ intersect at $X$ and $Y$. $O$ is an arbitrary point on the line $XY$ but not on $AD$. $CO$ intersects the circle with diameter $AC$ again at $M$, and $BO$ intersects the other circle again at $N$. Prove that the lines $AM,DN$, and $XY$ are concurrent.
by B. Mihailov
1995 IMO Shortlist G2 (GER)Let $A,B$, and $C$ be noncollinear points. Prove that there is a unique point $X$ in the plane of $ABC$ such that $XA^2+XB^2+AB^2 = XB^2+XC^2+BC^2 = XC^2+XA^2+CA^2$.
by Arthur Engel
1995 IMO Shortlist G3 (TUR)The incircle of $ABC$ touches $BC, CA$, and $AB$ at $D, E$, and $F$ respectively. $X$ is a point inside $ABC$ such that the incircle of $XBC$ touches $BC$ at $D$ also, and touches $CX$ and $XB$ at $Y$ and $Z$, respectively. Prove that $EFZY$ is a cyclic quadrilateral.
1995 IMO Shortlist G4 (UKR)
An acute triangle $ABC$ is given. Points $A_1$ and $A_2$ are taken on the side $BC$ (with $A_2$ between $A_1$ and $C$), $B_1$ and $B_2$ on the side AC (with B_2 between $B_1$ and $A$), and $C_1$ and $C_2$ on the side $AB$ (with $C_2$ between $C_1$ and $B$) such that $\angle AA_1A_2 = \angle AA_2A_1 = \angle BB_1B_2 = \angle BB_2B_1 = \angle CC_1C_2 = \angle CC_2C_1$. The lines $AA_1,BB_1$, and $CC_1$ form a triangle, and the lines $AA_2,BB_2$, and $CC_2$ form a second triangle. Prove that all six vertices of these two triangles lie on a single circle.
by Vyacheslav Yasinskiy
1995 IMO Shortlist G5 (NZL) problem 5Let $ABCDEF$ be a convex hexagon with $AB = BC =CD, DE = EF = FA$, and $ \angle BCD = \angle EFA = \pi /3$ (that is, $60^o$). Let $G$ and $H$ be two points interior to the hexagon such that angles $AGB$ and $DHE$ are both $2\pi /3$ (that is,$120^o$). Prove that $AG+GB+GH+DH +HE \ge CF$.
by A. McNaughton
1995 IMO Shortlist G6 (USA)Let $A_1A_2A_3A_4$ be a tetrahedron, $G$ its centroid, and $A_1',A_2',A_3'$, and $A_4'$ the points where the circumsphere of $A_1A_2A_3A_4$ intersects $GA_1,GA_2,GA_3$, and $GA_4$, respectively. Prove that $GA_1 \cdot GA_2 \cdot GA_3 \cdot GA_4 \le GA_1' \cdot GA_2' \cdot GA_3' \cdot GA_4'$ and $\frac{1}{GA_1'}+\frac{1}{GA_2'}+\frac{1}{GA_3'} +\frac{1}{GA_4' }\le \frac{1}{GA_1}+\frac{1}{GA_2}+\frac{1}{GA_3}+\frac{1}{GA_4}$.
by Titu Andreescu
1995 IMO Shortlist G7 (LVA)$O$ is a point inside a convex quadrilateral $ABCD$ of area $S$. $K, L, M$, and $N$ are interior points of the sides $AB, BC, CD$, and $DA$ respectively. If $OKBL$ and $OMDN$ are parallelograms, prove that $\sqrt{S} \ge \sqrt{S_1} +\sqrt{S_2}$, where $S_1$ and $S_2$ are the areas of $ONAK$ and $OLCM$ respectively.
1995 IMO Shortlist G8 (COL)
Let $ABC$ be a triangle. A circle passing through $B$ and $C$ intersects the sides $AB$ and $AC$ again at $C'$ and $B'$, respectively. Prove that $BB',CC'$, and $HH'$ are concurrent, where $H$ and $ H'$ are the orthocenters of triangles $ABC$ and $AB'C'$ respectively.
by Germán Rincón
1996 IMO Shortlist G1 (UNK)
Let triangle $ABC$ have orthocenter $H$, and let $P$ be a point on its circumcircle, distinct from $A,B,C$. Let $E$ be the foot of the altitude $BH$, let $PAQB$ and $PARC$ be parallelograms, and let $AQ$ meet HR in $X$. Prove that $EX$ is parallel to $AP$.
by David Monk
1996 IMO Shortlist G2 (CAN) problem 2
Let $P$ be a point inside △ABC such that $\angle APB - \angle C = \angle APC- \angle B$. Let $D,E$ be the incenters of $\triangle APB,\triangle APC$ respectively. Show that $AP,BD$ and $CE$ meet in a point.
by J.P. Grossman
1996 IMO Shortlist G3 (UNK)
Let $ABC$ be an acute-angled triangle with $BC> CA$. Let $O$ be the circumcenter, $H$ its orthocenter, and $F$ the foot of its altitude $CH$. Let the perpendicular to $OF$ at $F$ meet the side $CA$ at $P$. Prove that $\angle FHP = \angle BAC$.
Possible second part: What happens if $|BC|\le |CA|$ (the triangle still being acuteangled)?
by David Monk
1996 IMO Shortlist G4 (USA)
Let $\triangle ABC$ be an equilateral triangle and let P be a point in its interior. Let the lines $AP,BP,CP$ meet the sides $BC,CA,AB$ in the points $A_1,B_1,C_1$ respectively. Prove that $A_1B_1 \cdot B_1C_1\cdot C_1A_1 \ge A_1B \cdot B_1C \cdot C_1A$.
by Titu Andreescu
1996 IMO Shortlist G5 (ARM) problem 5
Let $ABCDEF$ be a convex hexagon such that $AB$ is parallel to $DE, BC$ is parallel to $EF$, and $CD$ is parallel to $AF$. Let $R_A,R_C,R_E$ be the circumradii of triangles $FAB,BCD,DEF$ respectively, and let $P$ denote the perimeter of the hexagon. Prove that $R_A+R_C+R_E \ge \frac{P}{2}$.
by N.M. Sedrakyan
1996 IMO Shortlist G6 (ARM)
Let the sides of two rectangles be $\{a,b\}$ and $\{c,d\}$ with $a ,c \le d < b$ and $ab < cd$. Prove that the first rectangle can be placed within the second one if and only if $(b^2-a^2)^2 \le (bd-ac)^2+(bc-ad)^2$.
1996 IMO Shortlist G7 (UNK)
Let $ABC$ be an acute-angled triangle with circumcenter $O$ and circumradius $R$. Let $AO$ meet the circle $BOC$ again in $A'$, let BO meet the circle $COA$ again in $B'$, and let $CO$ meet the circle $AOB$ again in $C'$. Prove that $OA' \cdot OB' \cdot OC' \ge 8R^3$. When does equality hold?
by Christopher Bradley
1996 IMO Shortlist G8 (RUS)
Let $ABCD$ be a convex quadrilateral, and let $RA, RB, RC$, and $RD$ denote the circumradii of the triangles $DAB, ABC, BCD$, and $CDA$ respectively. Prove that $RA+RC \ge RB+RD$ if and only if $\angle A+ \angle C > \angle B+\angle D$.
1996 IMO Shortlist G9 (UKR)
In the plane are given a point $O$ and a polygon $F$ (not necessarily convex). Let $P$ denote the perimeter of $F, D$ the sum of the distances from $O$ to the vertices of $F$, and $H$ the sum of the distances from O to the lines containing the sides of $F$. Prove that $D^2- H^2 \ge \frac{P^2}{4}$ .
by Vyacheslav Yasinskiy
1997 IMO Shortlist ISL5 (ROU)
Let $ABCD$ be a regular tetrahedron and $M,N$ distinct points in the planes $ABC$ and $ADC$ respectively. Show that the segments $MN,BN,MD$ are the sides of a triangle.
by Mircea Becheanu
1997 IMO Shortlist ISL7 (RUS)
Let $ABCDEF$ be a convex hexagon such that $AB =BC, CD= DE, EF = FA$. Prove that $\frac{BC}{ BE}+ \frac{DE}{DA} + \frac{FA}{ FC} \ge \frac{3}{2}$. When does equality occur?
by Valentina Kirichenko
1997 IMO Shortlist ISL8 (UNK) problem 2
In triangle $ABC$ the angle at $A $ is the smallest. A line through $A$ meets the circumcircle again at the point $U$ lying on the arc $BC$ opposite to $A$.
The perpendicular bisectors of $CA$ and $AB$ meet $AU$ at $V$ and $W$, respectively, and the lines $CV,BW$ meet at $T$. Show that $AU = TB+TC$.
Original formulation.
Four different points $A,B,C,D$ are chosen on a circle $G$ such that the triangle $BCD$ is not right-angled. Prove that:
a) The perpendicular bisectors of $AB$ and $AC$ meet the line $AD$ at certain points $W$ and $V$, respectively, and that the lines $CV$ and $BW$ meet at a certain point $T$.
b) The length of one of the line segments $AD, BT$, and $CT$ is the sum of the lengths of the other two.
by David Monk
1997 IMO Shortlist ISL9 (USA)
Let $A_1A_2A_3$ be a nonisosceles triangle with incenter $I$. Let $C_i, i = 1,2,3,$ be the smaller circle through $I$ tangent to $A_iA_{i+1}$ and $A_iA_{i+2}$ (the addition of indices being mod $3$). Let $B_i, i = 1,2,3$, be the second point of intersection of $C_{i+1}$ and $C_{i+2}$. Prove that the circumcenters of the triangles $A_1B_1I,A_2B_2I,A_3B_3I$ are collinear.
by T. Andreescu & K. Kedlaya
1997 IMO Shortlist ISL16 (BLR)
In an acute-angled triangle $ABC$, let $AD,BE$ be altitudes and $AP,BQ$ internal bisectors. Denote by $I$ and $O$ the incenter and the circumcenter of the triangle, respectively. Prove that the points $D, E$, and $I$ are collinear if and only if the points $P, Q$, and $O$ are collinear.
by Igor Voronovich
1997 IMO Shortlist ISL18 (UNK)
The altitudes through the vertices $A,B,C$ of an acute-angled triangle $ABC$ meet the opposite sides at $D,E,F$, respectively. The line through $D$ parallel to $EF$ meets the lines $AC$ and $AB$ at $Q$ and $R$, respectively. The line $EF$ meets $BC$ at $P$. Prove that the circumcircle of the triangle $PQR$ passes through the midpoint of $BC$.
by David Monk
1997 IMO Shortlist ISL20 (IRL)
Let D be an internal point on the side $BC$ of a triangle $ABC$. The line $AD$ meets the circumcircle of $ABC$ again at $X$. Let $P$ and $Q$ be the feet of the perpendiculars from $X$ to $AB$ and $AC$, respectively, and let $\gamma$ be the circle with diameter $XD$. Prove that the line $PQ$ is tangent to $\gamma$ if and only if $AB = AC$.
by Kevin Hutchinson
1997 IMO Shortlist ISL23 (UNK)
Let $ABCD$ be a convex quadrilateral and $O$ the intersection of its diagonals $AC$ and $BD$. If $OA sin \angle A + OC sin \angle C = OB sin\angle B+ OD sin \angle D$, prove that $ABCD$ is cyclic.
by Christopher Bradley
1997 IMO Shortlist ISL25 (POL)
The bisectors of angles $A,B,C$ of a triangle $ABC$ meet its circumcircle again at the points $K,L,M$, respectively. Let $R$ be an internal point on the side $AB$. The points $P$ and $Q$ are defined by the following conditions: $RP$ is parallel to $AK$, and $BP$ is perpendicular to $BL$, $RQ$ is parallel to $BL$, and $AQ$ is perpendicular to $AK$. Show that the lines $KP,LQ,MR$ have a point in common.
by Marcin Kuczma
1998 IMO Shortlist ISL1 (LUX) problem 1
A convex quadrilateral $ABCD$ has perpendicular diagonals. The perpendicular bisectors of $AB$ and $CD$ meet at a unique point $P$ inside $ABCD$. Prove that $ABCD$ is cyclic if and only if triangles $ABP$ and $CDP$ have equal areas.
by Charles Leytem
1998 IMO Shortlist ISL2 (POL)
Let $ABCD$ be a cyclic quadrilateral. Let $E$ and $F$ be variable points on the sides $AB$ and $CD$, respectively, such that $AE : EB =CF : FD$. Let $P$ be the point on the segment $EF$ such that $PE : PF = AB : CD$. Prove that the ratio between the areas of triangles $APD$ and $BPC$ does not depend on the choice of $E$ and $F$.
by Waldemar Pompe
1998 IMO Shortlist ISL3 (UKR) problem 5
Let $I$ be the incenter of triangle $ABC$. Let $K, L$, and $M$ be the points of tangency of the incircle of $ABC$ with $AB, BC$, and $CA$, respectively. The line $t$ passes through $B$ and is parallel to $KL$. The lines $MK$ and $ML$ intersect $t$ at the points $R$ and $S$. Prove that $\angle RIS$ is acute.
by V. Yasinskiy
1998 IMO Shortlist ISL4 (ARM)
Let $M$ and $N$ be points inside triangle $ABC$ such that $\angle MAB = \angle NAC$ and $\angle MBA = \angle NBC$. Prove that $\frac{AM\cdot AN }{AB\cdot AC} + \frac{BM \cdot BN }{BA\cdot BC} + \frac{CM\cdot CN }{CA\cdot CB} = 1$.
1998 IMO Shortlist ISL5 (FRA)
Let $ABC$ be a triangle, $H$ its orthocenter, $O$ its circumcenter, and $R$ its circumradius. Let $D$ be the reflection of $A$ across $BC$, $E$ that of $B$ across $CA$, and $F$ that of $C$ across $AB$. Prove that $D, E$, and $F$ are collinear if and only if $OH = 2R$.
1998 IMO Shortlist ISL6 (POL)
Let $ABCDEF$ be a convex hexagon such that $\angle B+\angle D+\angle F = 360^o$ and $\frac{AB}{ BC} \cdot \frac{CD}{ DE }\cdot \frac{EF}{ FA} = 1$. Prove that $\frac{BC}{ CA} \cdot \frac{AE}{ EF} \cdot \frac{FD}{ DB} = 1$.
by Waldemar Pompe
1998 IMO Shortlist ISL7 (UNK)
Let ABC be a triangle such that $\angle ACB = 2 \angle ABC$. Let $D$ be the point on the side $BC$ such that $CD = 2BD$. The segment $AD$ is extended to $E$ so that $AD = DE$. Prove that $\angle ECB+180^o = 2 \angle EBC$.
by David Monk
1998 IMO Shortlist ISL8 (IND)
Let ABC be a triangle such that $\angle A = 90^o $ and $\angle B < \angle C$. The tangent at $A$ to its circumcircle $\omega$ meets the line $BC$ at $D$. Let $E$ be the reflection of $A$ across $BC$, $X$ the foot of the perpendicular from $A$ to $BE$, and $Y$ the midpoint of $AX$. Let the line $BY$ meet $\omega$ again at $Z$. Prove that the line $BD$ is tangent to the circumcircle of triangle $ADZ$.
by Sambuddha Roy
1999 IMO Shortlist G1 (ARM)
Let $ABC$ be a triangle and $M$ an interior point. Prove that $min\{MA,MB,MC\}+MA+MB+MC <AB+AC+BC$.
A circle is called a separator for a set of five points in a plane if it passes through three of these points, it contains a fourth point in its interior, and the fifth point is outside the circle. Prove that every set of five points such that no three are collinear and no four are concyclic has exactly four separators.
A set $S$ of points in space will be called completely symmetric if it has at least three elements and satisfies the following condition: For every two distinct points $A,B$ from S the perpendicular bisector of the segment $AB$ is an axis of symmetry for $S$. Prove that if a completely symmetric set is finite, then it consists of the vertices of either a regular polygon, a regular tetrahedron, or a regular octahedron.
For a triangle $T = ABC$ we take the point $X$ on the side $(AB)$ such that $AX/XB = 4/5$, the point $Y $von the segment $(CX) $ such that $CY = 2YX$, and, if possible, the point $Z$ on the ray $(CA$ such that $\angle CXZ = 180^o - \angle ABC$. We denote by $S$ the set of all triangles $T$ for which $\angle XYZ = 45^o$. Prove that all the triangles from $S$ are similar and find the measure of their smallest angle.
Let $ABC$ be a triangle, $\Omega$ its incircle and $\Omega_a,\Omega_b,\Omega_c$ three circles orthogonal to $\Omega$ passing through $B$ and $C$, $A$ and $C$, and $A$ and $B$ respectively. The circles $\Omega_a,\Omega_b$ meet again in $C'$, in the same way we obtain the points $B'$ and $A'$. Prove that the radius of the circumcircle of $A'B'C'$ is half the radius of $\omega$.
1999 IMO Shortlist G6 (RUS) problem 5
Two circles $\Omega_1$ and $\Omega_2$ touch internally the circle $\Omega$ in $M$ and $N$, and the center of $\Omega_2$ is on $\Omega_1$. The common chord of the circles $\Omega_1$ and $\Omega_2$ intersects $\Omega$ in $A$ and $B$. $MA$ and $MB$ intersect $\Omega_1$ in $C$ and $D$. Prove that $\Omega_2$ is tangent to $CD$.
The point $M$ inside the convex quadrilateral $ABCD$ is such that $MA = MC$, $\angle AMB = \angle MAD+\angle MKD$, $\angle CMD = \angle MCB+ \angle MAB$. Prove that $AB\cdot CM = BC\cdot MD$ and $BM\cdot AD = MA\cdot CD$.
1999 IMO Shortlist G8 (RUS)
Points $A,B,C$ divide the circumcircle $\Omega$ of the triangle $ABC$ into three arcs. Let $X$ be a variable point on the arc $AB$, and let $O_1,O_2$ be the incenters of the triangles $CAX$ and $CBX$. Prove hat the circumcircle of the triangle $XO_1O_2$ intersects $\Omega$ in a fixed point.
2000 IMO Shortlist G1 (NLD)
Let $ABC$ be a triangle and $M$ an interior point. Prove that $min\{MA,MB,MC\}+MA+MB+MC <AB+AC+BC$.
by Nairi M. Sedrakyan
1999 IMO Shortlist G2 (JPN)A circle is called a separator for a set of five points in a plane if it passes through three of these points, it contains a fourth point in its interior, and the fifth point is outside the circle. Prove that every set of five points such that no three are collinear and no four are concyclic has exactly four separators.
by Shin Hitotsumatsu
1999 IMO Shortlist G3 (EST) problem 1A set $S$ of points in space will be called completely symmetric if it has at least three elements and satisfies the following condition: For every two distinct points $A,B$ from S the perpendicular bisector of the segment $AB$ is an axis of symmetry for $S$. Prove that if a completely symmetric set is finite, then it consists of the vertices of either a regular polygon, a regular tetrahedron, or a regular octahedron.
by Jan Villemson
1999 IMO Shortlist G4 (UNK)For a triangle $T = ABC$ we take the point $X$ on the side $(AB)$ such that $AX/XB = 4/5$, the point $Y $von the segment $(CX) $ such that $CY = 2YX$, and, if possible, the point $Z$ on the ray $(CA$ such that $\angle CXZ = 180^o - \angle ABC$. We denote by $S$ the set of all triangles $T$ for which $\angle XYZ = 45^o$. Prove that all the triangles from $S$ are similar and find the measure of their smallest angle.
by David Monk
1999 IMO Shortlist G5 (FRA)Let $ABC$ be a triangle, $\Omega$ its incircle and $\Omega_a,\Omega_b,\Omega_c$ three circles orthogonal to $\Omega$ passing through $B$ and $C$, $A$ and $C$, and $A$ and $B$ respectively. The circles $\Omega_a,\Omega_b$ meet again in $C'$, in the same way we obtain the points $B'$ and $A'$. Prove that the radius of the circumcircle of $A'B'C'$ is half the radius of $\omega$.
1999 IMO Shortlist G6 (RUS) problem 5
Two circles $\Omega_1$ and $\Omega_2$ touch internally the circle $\Omega$ in $M$ and $N$, and the center of $\Omega_2$ is on $\Omega_1$. The common chord of the circles $\Omega_1$ and $\Omega_2$ intersects $\Omega$ in $A$ and $B$. $MA$ and $MB$ intersect $\Omega_1$ in $C$ and $D$. Prove that $\Omega_2$ is tangent to $CD$.
by P. Kozhevnikov
1999 IMO Shortlist G7 (ARM)The point $M$ inside the convex quadrilateral $ABCD$ is such that $MA = MC$, $\angle AMB = \angle MAD+\angle MKD$, $\angle CMD = \angle MCB+ \angle MAB$. Prove that $AB\cdot CM = BC\cdot MD$ and $BM\cdot AD = MA\cdot CD$.
1999 IMO Shortlist G8 (RUS)
Points $A,B,C$ divide the circumcircle $\Omega$ of the triangle $ABC$ into three arcs. Let $X$ be a variable point on the arc $AB$, and let $O_1,O_2$ be the incenters of the triangles $CAX$ and $CBX$. Prove hat the circumcircle of the triangle $XO_1O_2$ intersects $\Omega$ in a fixed point.
2000 IMO Shortlist G1 (NLD)
In the plane we are given two circles intersecting at $X$ and $Y$. Prove that there exist four points $A,B,C,D$ with the following property:
For every circle touching the two given circles at $A$ and $B$, and meeting the line $XY$ at $C$ and $D$, each of the lines $AC,AD,BC,BD$ passes through one of these points.
2000 IMO Shortlist G2 (RUS) problem 1
Two circles $G_1$ and $G_2$ intersect at $M$ and $N$. Let $AB$ be the line tangent to these circles at $A$ and $B$, respectively, such that $M$ lies closer to $AB$ than N. Let $CD$ be the line parallel to $AB$ and passing through $M$, with $C$ on $G_1$ and $D$ on $G_2$. Lines $AC$ and $BD$ meet at $E$, lines $AN$ and $CD$ meet at $P$; lines $BN$ and $CD$ meet at $Q$. Show that $EP = EQ$.
by Sergey Berlov
2000 IMO Shortlist G3 (IND)
Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$. Show that there exist points $D, E$, and $F$ on sides $BC, CA$, and $AB$ respectively such that $OD+DH = OE +EH = OF +FH$ and the lines $AD, BE$, and $CF$ are concurrent.
by C.R. Pranesachar
2000 IMO Shortlist G4 (RUS)
Let $A_1A_2...A_n$ be a convex polygon, $ n \ge 4$. Prove that $A_1A_2... A_n$ is cyclic if and only if to each vertex $A_j$ one can assign a pair $(b_j ,c_j)$ of real numbers, $j = 1,2,...n$, such that $A_iA_j = b_jc_i- b_ic_j$ for all $i, j$ with $1 \le i \le j \le n$.
2000 IMO Shortlist G5 (UNK)
The tangents at $B$ and $A$ to the circumcircle of an acute-angled triangle $ABC$ meet the tangent at $C$ at $T$ and $U$ respectively. $AT$ meets $BC$ at $P$, and $Q$ is the midpoint of $AP$, BU meets $CA$ at $R$, and $S$ is the midpoint of $BR$. Prove that $\angle ABQ = \angle BAS$. Determine, in terms of ratios of side lengths, the triangles for which this angle is a maximum.
by David Monk
2000 IMO Shortlist G6 (ARG)
Let $ABCD$ be a convex quadrilateral with $AB$ not parallel to $CD$, let $X$ be a point inside $ABCD$ such that $\angle ADX = \angle BCX < 90^o$ and $\angle DAX = \angle CBX < 90^o$. If $Y$ is the point of intersection of the perpendicular bisectors of $AB$ and $CD$, prove that $\angle AYB = 2 \angle ADX$.
2000 IMO Shortlist G7 (IRN)
Ten gangsters are standing on a flat surface, and the distances between themare all distinct. At twelve o’clock, when the church bells start chiming, each of them fatally shoots the one among the other nine gangsters who is the nearest. At least how many gangsters will be killed?
2000 IMO Shortlist G8 (RUS) problem 6
$A_1A_2A_3$ is an acute-angled triangle. The foot of the altitude from $A_i$ is $K_i$, and the incircle touches the side opposite $A_i $ at $L_i$. The line $K_1K_2$ is reflected in the line $L_1L_2$. Similarly, the line K2K3 is reflected in $L_2L_3$, and $K_3K_1$ is reflected in $L_3L_1$. Show that the three new lines form a triangle with vertices on the incircle.
by L. Emelyanov, T. Emelyanova
Let $A_1$ be the
center of the square inscribed in acute triangle $ABC$ with two vertices of the
square on side $BC$. Thus one of the two remaining vertices of the square is on
side $AB$ and the other is on $AC$. Points $B_1, C_1$ are defined in a similar
way for inscribed squares with two vertices on sides $AC$ and $AB$,
respectively. Prove that lines $AA_1, BB_1, CC_1$ are concurrent.
by Vyacheslav Yasinskiy
2001 IMO
Shortlist G2 (KOR) problem 1
In acute
triangle $ABC$ with circumcenter $O$ and altitude $AP, \angle C \ge \angle B +
30^\circ$. Prove that $\angle A + \angle COP < 90^\circ$.
by Hojoo Lee
2001 IMO
Shortlist G3 (UNK)
Let $ABC$ be a
triangle with centroid $G$. Determine, with proof, the position of the point
$P$ in the plane of $ABC$ such that $AP \cdot AG+BP \cdot BG+CP \cdot CG$ is a
minimum, and express this minimum value in terms of the side lengths of $ABC$.
Let $M$ be a
point in the interior of triangle $ABC$. Let $A’$ lie on $BC$ with $MA’$
perpendicular to $BC$. Define $B’$ on $CA$ and $C’$ on $AB$ similarly. Define
$p(M) = \frac{MA’ \cdot MB’ \cdot MC’}{ MA \cdot MB \cdot MC} $. Determine,
with proof, the location of $M$ such that $p(M)$ is maximal. Let $\mu (ABC)$
denote this maximum value. For which triangles $ABC$ is the value of $\mu (ABC)$
maximal?
2001 IMO
Shortlist G5 (HEL)
Let $ABC$ be an
acute triangle. Let $DAC,EAB$, and $FBC$ be isosceles triangles exterior to
$ABC$, with$ DA = DC,EA = EB$, and $FB = FC$, such that $\angle ADC = 2\angle
BAC, \angle BEA = 2\angle ABC, \angle CFB = 2\angle ACB$. Let $D’$ be the
intersection of lines $DB$ and $EF$, let $E’$ be the intersection of $EC$ and
$DF$, and let $F’$ be the intersection of $FA$ and $DE$. Find, with proof, the
value of the sum $\frac{DB}{DD’} +\frac{EC}{EE’} +\frac{FA}{FF’}$ .
2001 IMO
Shortlist G6 (IND)
by Sotiris Louridas
Let $ABC$ be a
triangle and $P$ an exterior point in the plane of the triangle. Suppose
$AP,BP,CP$ meet the sides $BC,CA,AB$ (or extensions thereof) in $D,E, F$,
respectively. Suppose further that the areas of triangles $PBD, PCE, PAF$ are all
equal. Prove that each of these areas is equal to the area of triangle $ABC$
itself.
by C.R. Pranesachar
2001 IMO
Shortlist G7 (BGR)
Let $O$ be an
interior point of acute triangle $ABC$. Let $A_1$ lie on $BC$ with $OA_1$
perpendicular to $BC$. Define $B_1$ on $CA$ and $C_1$ on $AB$ similarly. Prove
that $O$ is the circumcenter of $ABC$ if and only if the perimeter of
$A_1B_1C_1$ is not less than any one of the perimeters of $AB_1C_1,BC_1A_1$,
and $CA_1B_1$.
by Emil Stoyanov
2001 IMO
Shortlist G8 (ISR) problem 5
Let $ABC$ be a
triangle with $\angle BAC = 60^\circ$. Let $AP$ bisect $\angle BAC$ and let
$BQ$ bisect $\angle ABC$, with $P$ on $BC$ and $Q$ on $AC$. If $AB +BP =
AQ+QB$, what are the angles of the triangle?
by Shay Gueron
2002 IMO
Shortlist G1 (FRA)
Let $B$ be a
point on a circle $S_1$, and let $A$ be a point distinct from $B$ on the
tangent at $B$ to $S_1$. Let $C$ be a point not on $S_1$ such that the line
segment $AC$ meets $S_1$ at two distinct points. Let $S_2$ be the circle
touching $AC$ at $C$ and touching $S_1$ at a point $D$ on the opposite side of
$AC$ from $B$. Prove that the circumcentre of triangle $BCD$ lies on the
circumcircle of triangle $ABC$.
2002 IMO
Shortlist G2 (KOR)
Let $ABC$ be a
triangle for which there exists an interior point $F$ such that $ \angle
AFB =
\angle BFC = \angle CFA$. Let the lines $BF$ and $CF$ meet
the sides $AC$ and $AB$ at $D$ and $E$ respectively. Prove that $AB + AC
\ge 4 DE$.
by Hojoo Lee
2002 IMO
Shortlist G3 (KOR) problem 2
The circle $S$
has centre $O$, and $BC$ is a diameter of $S$. Let $A$ be a point of $S$ such
that $\angle AOB < 120^\circ$. Let $D$ be the midpoint of the arc $AB$ which
does not contain $C$. The line through
$O$ parallel to $DA$ meets the line $AC$ at I. The perpendicular bisector of
$OA$ meets $S$ at $E$ and at $F$. Prove that $I$ is the incentre of the
triangle $CEF$.
by Hojoo Lee
2002 IMO Shortlist
G4 (RUS)
Circles $S_1$
and $S_2$ intersect at points $P$ and $Q$. Distinct points $A_1$ and $B_1$ (not
at $P$ or $Q$) are selected on $S_1$. The lines $A_1P$ and $B_1P$ meet $S_2$
again at $A_2$ and $B_2$ respectively, and the lines $A_1B_1$ and $A_2B_2$ meet
at $C$. Prove that, as $A_1$ and $B_1$ vary, the circumcentres of triangles
$A_1A_2C$ all lie on one fixed circle.
2002 IMO
Shortlist G5 (AUS)
For any set $S$
of five points in the plane, no three of which are collinear, let $M (S)$ and
$m (S)$ denote the greatest and smallest areas, respectively, of triangles
determined by three points from $S$. What is the minimum possible value of $M
(S) / m (S)$?
by Angelo Di Pasquale
2002 IMO
Shortlist G6 (UKR) problem 6
Let $n \ge 3$ be a positive integer. Let $C_1, C_2,
C_3, … , C_n$ be unit circles in the plane, with centres $O_1, O_2, O_3, … ,
O_n$ respectively. If no line meets more than two of the circles, prove that $
\sum_{1\le I < j \le n} \frac{1}{O_ i O_ j} \le frac{(n- 1) \pi}{4}$.
by V. Yasinskiy
2002 IMO
Shortlist G7 (BGR)
The incircle $\Omega$ of the acute-angled triangle $ABC $ is tangent to $BC$ at $K$. Let $AD$
be an altitude of triangle $ABC$ and let $M$ be the midpoint of $AD$. If $N$ is
the other common point of and $KM$, prove that $\Omega$ and the circumcircle of
triangle $BCN$ are tangent at $N$.
2002 IMO
Shortlist G8 (ARM)
Let $S_1$ and
$S_2$ be circles meeting at the points $A$ and $B$. A line through $A$ meets
$S_1$ at C and $S_2$ at $D$. Points $M, N, K$ lie on the line segments $CD, BC,
BD$ respectively, with $MN$ parallel to $BD$ and $MK$ parallel to $BC$. Let $E$
and $F$ be points on those arcs $BC$ of $S_1$ and $BD$ of $S_2$ respectively
that do not contain $A$. Given that $EN$ is perpendicular to $BC$ and $FK$ is
perpendicular to $BD$ prove that $\angle EMF = 90^\circ$ .
2003 IMO Shortlist
G1 (FIN) problem 4
Let $ABCD$ be a cyclic
quadrilateral. Let $P , Q, R$ be the
feet of the perpendiculars from $D$ to the lines $BC, CA, AB$, respectively.
Show that $PQ = QR$ if and only if the bisectors of $\angle ABC$ and $\angle
ADC$ are concurrent with $AC$.
by Matti Lehtinen
2003 IMO Shortlist
G2 (HEL)
Three distinct points $A, B, C$ are
fixed on a line in this order. Let $ \Gamma$ be a circle passing through $A$
and $C$ whose centre does not lie on the line $AC$. Denote by $P$ the intersection of the
tangents to $\Gamma$ at $A$ and
$C$. Suppose $\Gamma$ meets the segment
$PB$ at $Q$. Prove that the intersection
of the bisector of $\angle AQC$ and the line $AC$ does not depend on the choice
of $\Gamma$.
2003 IMO
Shortlist G3 (IND)
Let $ABC$ be a triangle and let $P$
be a point in its interior. Denote by $D, E, F$ the feet of the perpendiculars
from $P$ to the lines $BC, CA, AB$, respectively. Suppose that
$AP^2 + PD^2 = BP^2 + PE^2 = CP^2 +
PF^2$.
Denote by $I_A, I_B , I_C$ the
excentres of the triangle $ABC$. Prove that $P$ is the circumcentre of the
triangle $I_AI_BI_C$ .
by C.R. Pranesachar
2003 IMO
Shortlist G4 (ARM)
Let $\Gamma_1, \Gamma_2, \Gamma_3,
\Gamma_4$ be distinct circles such that $\Gamma_1, \Gamma_3$ are externally
tangent at $P$ , and $\Gamma_2, \Gamma_4$ are externally tangent at the same
point $P$ . Suppose that $\Gamma_1$ and $ \Gamma_2, \Gamma_2$ and $\Gamma_3,
\Gamma_3$ and $ \Gamma_4, \Gamma_4$ and $\Gamma_1$ meet at $A, B, C, D$,
respectively, and that all these points are different from $P$ . Prove that
$\frac{AB \cdot BC}{AD \cdot
DC}=\frac{PB^2}{PD^2}$.
2003 IMO
Shortlist G5 (KOR) (also here)
Let $ABC$ be an isosceles triangle with $AC = BC$,
whose incentre is $I$. Let $P$ be a point on the circumcircle of the
triangle $AIB$ lying inside the triangle $ABC$. The lines through $P$ parallel
to $CA$ and $CB$ meet $AB$ at $D$ and $E$, respectively. The line through $P$
parallel to $AB$ meets $CA$ and $CB$ at $F$
and $G$, respectively. Prove that
the lines $DF$ and $EG$ intersect on the
circumcircle of the triangle $ABC$.
by Hojoo Lee
2003 IMO Shortlist
G6 (POL) problem 3
Each pair of opposite sides of a
convex hexagon has the following property:
the distance between their midpoints
is equal to $ \sqrt{3} / 2 $ times the sum of their lengths. Prove that all the
angles of the hexagon are equal.
by Waldemar Pompe
2003 IMO
Shortlist G7 (SAF)
Let $ABC$ be a triangle with
semiperimeter $s$ and inradius $r$. The
semicircles with diameters $BC, CA, AB$ are drawn on the outside of the
triangle $ABC$. The circle tangent to all three semicircles has radius $t$.
Prove that $ \frac{s}{2}< t \le \frac{s}{2}
+ \Big(1 -\frac{\sqrt{3}}{2}\Big) r $.
2004 IMO Shortlist G1 (ROU) problem 1
by Dirk Laurie
Let $ABC$ be an acute-angled triangle with $AB \ne AC$. The circle with diameter $BC$ intersects the sides $AB$ and $AC$ at $M$ and $N$ , respectively. Denote by $O$ the midpoint of $BC$. The bisectors of the angles $BAC$ and $MON$ intersect at $R$. Prove that the circumcircles of the triangles $BMR$ and $CNR$ have a common point lying on the line segment $BC$.
by D. Serbanescu & V. Vornicu
2004 IMO Shortlist G2 (KAZ)
The circle $\Gamma$ and the line $l$
do not intersect. Let $AB$ be the diameter of \Gamma perpendicular to $l$, with
$B$ closer to $l$ than $A$. An arbitrary point $C \ne A, B$ is chosen on
$\Gamma$. The line $AC$ intersects $l$
at $D$. The line $DE$ is tangent to $\Gamma$ at $E$, with $B$ and $E$ on
the same side of $AC$. Let $BE$ intersect $l$ at $F$ , and let $AF $ intersect
$ \Gamma$ at $G \ne A$. Prove that the reflection of $G$ in $AB$ lies on the
line $CF$ .
2004 IMO Shortlist G3 (KOR)
Let $O$ be the circumcenter of an
acute-angled triangle $ABC$ with $\angle
B < \angle C$. The line $AO$ meets the side $BC$ at $D$. The
circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$ , respectively.
Extend the sides $BA$ and $ CA$ beyond $A$, and choose on the respective
extensions points $G$ and $H$ such that $AG = AC$ and $AH = AB$. Prove that the quadrilateral $EFGH$ is a
rectangle if and only if $\angle ACB -
\angle ABC = 60^\circ$.
by Hojoo Lee
In a convex quadrilateral $ABCD$ the
diagonal $BD$ does not bisect the angles $ABC$ and $CDA$. The point $P$
lies inside $ABCD$ and satisfies $\angle
PBC = \angle DBA$ and
$\angle PDC = \angle BDA$. Prove that $ABCD$ is a cyclic
quadrilateral if and only if $AP = CP$ .
2004 IMO Shortlist G5 (SCG)
by Waldemar Pompe
Let $A_1A_2 … A_n$ be a regular
$n$-gon. The points $B_1,..., B_{n-1}$ are defined as follows:
If $i =1$ or $ i = n – 1$, then $B_i
$ is the midpoint of the side $A_iA_{i+1}$,
If $i \ne 1, i \ne n – 1$ and $S$ is
the intersection point of $A_iA_{i+1}$ and $A_nA_i$, then $B_i$ is the intersection point of the bisector of
the angle $A_iSA_{i+1}$ with $A_iA_{i+1}$.
Prove the equality $ \angle A_1B_1A_n + \angle A_1B_2A_n + … +\angle A_1B_{n-1}A_n = 180^\circ$.
2004 IMO Shortlist G6 (UNK)
by Dušan Ðukic
Let $P$ be a convex polygon. Prove
that there is a convex hexagon which is contained in $P $and which occupies at
least $75$ percent of the area of $P$.
2004 IMO Shortlist G7 (RUS)
by Ben Green & Edward Crane
For a given triangle $ABC$, let $X$
be a variable point on the line $BC$ such that $C$ lies between $B$ and $X$ and
the incircles of the triangles $ABX$ and $ACX$ intersect at two distinct points
$P$ and $Q$. Prove that the line $PQ$
passes through a point independent of $X$.
2004 IMO Shortlist G8 (SCG)
A cyclic quadrilateral $ABCD$ is
given. The lines $AD$ and $BC$ intersect at $E$, with $C$ between $B$ and $E$;
the diagonals $AC$ and $BD$ intersect at $F$ . Let $M$ be the midpoint of the
side $CD$, and let $N \ne M$ be a point on the circumcircle of the triangle
$ABM$ such that $AN/BN = AM/BM$ . Prove
that the points $E, F$ and $N$ are
collinear.
by Dušan Ðukic
2005 IMO Shortlist G1 (HEL)
In a triangle $ABC$ satisfying $AB +
BC = 3AC$ the incircle has centre $I$ and touches the sides $AB$ and $BC$ at
$D$ and $E$, respectively. Let $K$ and $L$ be the symmetric points of $D$ and
$E$ with respect to $ I$. Prove that the quadrilateral $ACKL$ is cyclic.
by Dimitris Kontogiannis
2005 IMO Shortlist G2 (ROU) problem 1
Six points are chosen on the sides
of an equilateral triangle $ABC$: $A_1, A_2$ on $BC, B_1, B_2$ on $CA$, and
$C_1, C_2$ on $AB$, so that they are the vertices of a convex hexagon
$A_1A_2B_1B_2C_1C_2$ with equal side lengths. Prove that the lines $A_1B_2,
B_1C_2$ and $C_1A_2$ are concurrent.
by Bogdan Enescu
2005 IMO Shortlist G3 (UKR)
Let $ABCD$ be a parallelogram. A
variable line $l$ passing through the point $A$ intersects the rays $BC$ and
$DC$ at points $X$ and $Y$ , respectively. Let $K$ and $L$ be the centres of
the excircles of triangles $ABX$ and $ADY$ , touching the sides $BX$ and $DY$ ,
respectively. Prove that the size of angle $KCL$ does not depend on the choice
of the line $ l$.
by Vyacheslav Yasinskiy
Let$ ABCD$ be a fixed convex
quadrilateral with $BC = DA$ and $BC$ not parallel to $DA$. Let two variable
points $E$ and $F$ lie on the sides $BC$ and $DA$, respectively, and satisfy
$BE = DF$ . The lines $AC$ and $BD$ meet at $P$ , the lines $BD$ and $EF$ meet
at $Q$, the lines $EF$ and $AC$ meet at $R$. Prove that the circumcircles of
triangles $PQR$, as $E$ and $F$ vary, have a common point other than $P$ .
by Waldemar Pompe
2005 IMO Shortlist G5 (ROU)
Let $ABC$ be an acute-angled
triangle with $AB \ne AC$, let $H$ be its orthocentre and $M $ the midpoint of
$BC$. Points $ D$ on $AB$ and $E$ on
$AC$ are such that $AE = AD$ and $D, H,
E$ are collinear. Prove that $HM$ is orthogonal to the common chord of the
circumcircles of triangles $ABC$ and $ADE$.
The median $AM$ of a triangle $ABC$
intersects its incircle $\omega$ at $K$ and $L$. The lines through $K$ and $L$
parallel to BC intersect $\omega$ again at $X$ and $Y$ . The lines $AX$ and $AY$ intersect $BC$
at $P $ and $Q$. Prove that $BP = CQ$.
2005 IMO Shortlist G7 (KOR)
In an acute triangle $ABC$, let $D,
E, F , P , Q, R$ be the feet of perpendiculars from $A, B, C, A, B, C$ to $BC, CA, AB, EF , F D, DE$, respectively.
Prove that $p(ABC)p(P QR) \ge p(DEF)^2$,
where $ p(T)$ denotes the perimeter of the triangle $T$ .
by Hojoo Lee
2006 IMO Shortlist G1 (KOR) problem 1
Let $ABC$ be a triangle with
incentre $I$. A point $P$ in the interior of the triangle satisfies $\angle PBA
+ \angle PCA = \angle PBC + \angle PCB$.
Show that $AP \ge AI$ and that
equality holds if and only if $P$ coincides with $I$.
by Hojoo Lee
2006 IMO Shortlist G2 (UKR)
Let $ABCD$ be a trapezoid with
parallel sides $AB > CD$. Points $K$
and $L$ lie on the line segments $AB$ and $CD$, respectively, so that $AK/KB =
DL/LC$. Suppose that there are points $P$ and $Q$ on the line segment $KL$
satisfying $\angle APB = \angle BCD$ and
$ \angle CQD = \angle ABC$. Prove that the points $P , Q, B$ and $C$ are concyclic.
by Vyacheslav Yasinskiy
2006 IMO Shortlist G3 (USA)
Let $ABCDE$ be a convex pentagon
such that $\angle BAC = \angle CAD = \angle DAE$ and
$\angle ABC = \angle ACD = \angle ADE$. The diagonals $BD$ and $CE$ meet
at $P$ . Prove that the line $AP$ bisects the side $CD$.
by Zuming Feng
2006 IMO Shortlist G4 (RUS)
A point $D$ is chosen on the side
$AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ $ in such a
way that $BD = BA$. The incircle of $ABC$ is tangent to $AB$ and $AC $ at
points $K$ and $L$, respectively. Let $J$ be the incentre of triangle $BCD$.
Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.
2006 IMO Shortlist G5 (HEL)
In triangle $ABC$, let $J$ be the
centre of the excircle tangent to side $BC$ at $A_1$ and to the extensions of sides $AC$ and $AB$
at $B_1$ and $C_1$, respectively. Suppose that the lines $A_1B_1$ and $AB$ are
perpendicular and intersect at $D$. Let $E$ be the foot of the perpendicular
from $C_1$ to line $DJ$ . Determine the angles $\angle BEA_1$ and $\angle AEB_1$.
by Dimitris Kontogiannis
2006 IMO Shortlist G6 (BRA)
Circles $\omega_1$ and $\omega _2$
with centres $O_1$ and $O_2$ are
externally tangent at point $D$ and internally tangent to a circle $\omega$ at
points $E$ and $F$ , respectively. Line $t$ is the common tangent of $\omega_1$ and $\omega_2$ at $D$. Let $AB$ be the diameter of $\omega$ perpendicular to $t$, so that $A, E$ and
$O_1$ are on the same side of $t$. Prove
that lines $AO_1, BO_2, EF$ and $t$ are concurrent.
2006 IMO Shortlist G7 (SVK)
In a triangle $ABC$, let $M_a, M_b,
M_c$ be respectively the midpoints of
the sides $BC, CA, AB$ and $T_a, T_b, T_c $ be the midpoints of the arcs $BC,
CA, AB $ of the circumcircle of $ABC$, not containing the opposite vertices.
For $ i \in \{a, b, c\}$, let $\omega_i $be the circle with $M_iT_i$ as
diameter. Let $p_i $ be the common external tangent to $\omega_j , \omega_k
(\{i, j, k\} = \{a, b, c\})$ such that $\omega_i$ lies on the opposite side of
$p_i$ than $\omega_j , \omega_k$ do. Prove that the lines $p_a, p_b, p_c$ form
a triangle similar to $ABC$ and find the ratio of similitude.
by Tomáš Jurík
2006 IMO Shortlist G8 (POL)
Let $ABCD$ be a convex
quadrilateral. A circle passing through
the points $A$ and $D$ and a circle passing through the points $B$ and $C$ are
externally tangent at a point P inside the quadrilateral. Suppose that $\angle
P AB + \angle PDC \le 90^\circ$ and $\angle PBA + \angle PCD \le 90^\circ$ .
Prove that $AB + CD \ge BC + AD$.
by Waldemar Pompe
2006 IMO Shortlist G9 (RUS)
Points $A_1,B_1, C_1$ are chosen on
the sides $BC, CA, AB$ of a triangle $ABC$, respectively. The circumcircles of
triangles $AB_1C_1, BC_1A_1, CA_1B_1$ intersect the circumcircle of triangle
ABC again at points $A_2, B_2, C_2$, respectively ($A_2 \ne A, B_2 \ne B, C_2
\ne C$). Points $A_3, B_3, C_3$ are
symmetric to $A_1, B_1, C_1$ with respect to the midpoints of the sides $BC,
CA, AB$ respectively. Prove that the triangles $A_2B_2C_2$ and $A_3B_3C_3$ are
similar.
2006 IMO Shortlist G10 (SRB) problem 6
To each side a of a convex polygon we assign the maximum area of a triangle contained in the polygon and having a as one of its sides. Show that the sum of the areas assigned to all sides of the polygon is not less than twice the area of the polygon.
by Dušan Ðukic
2007 IMO Shortlist G1 (CZE) problem 4In triangle $ABC$, the angle bisector at vertex $C$ intersects the circumcircle and the perpendicular bisectors of sides $BC$ and $CA$ at points $R, P$, and $Q$, respectively. The midpoints of $BC$ and $CA$ are $S$ and $T$, respectively. Prove that triangles $RQT$ and $RPS$ have the same area.
by Marek Pechal
2007 IMO Shortlist G2 (CAN)Given an isosceles triangle $ABC$ with $AB = AC$. The midpoint of side $BC$ is denoted by $M$. Let $X$ be a variable point on the shorter arc $ MA$ of the circumcircle of triangle $ABM$. Let $T$ be the point in the angle domain $BMA$, for which $ \angle TMX = 90^o$ and $TX = BX$. Prove that $ \angle MTB - \angle CTM$ does not depend on $X$.
by Farzin Barekat
2007 IMO Shortlist G3 (UKR)The diagonals of a trapezoid $ABCD$ intersect at point $P$. Point $Q$ lies between the parallel lines $BC$ and $AD$ such that $\angle AQD = \angle CQB$, and line $CD$ separates points $P$ and $Q$. Prove that $\angle BQP =\angle DAQ$.
by Vyacheslav Yasinskiy
2007 IMO Shortlist G4 (LUX) problem 2Consider five points $A, B, C, D, E$ such that $ABCD$ is a parallelogram and BCED is a cyclic quadrilateral. Let $\ell$ be a line passing through $A$, and let $\ell $ intersect segment $ DC$ and line $BC$ at points $F$ and $G$, respectively. Suppose that $EF = EG = EC$. Prove that $\ell$ is the bisector of angle $DAB$.
by Charles Leytem
2007 IMO Shortlist G5 (UNK)Let $ABC$ be a fixed triangle, and let $A_1, B_1, C_1$ be the midpoints of sides $BC, CA, AB$, respectively. Let $P$ be a variable point on the circumcircle. Let lines $PA_1, PB_1, PC_1$ meet the circumcircle again at $A', B', C' $ respectively. Assume that the points $A, B, C, A', B', C' $ are distinct, and lines $AA', BB', CC' $ form a triangle. Prove that the area of this triangle does not depend on $P$.
by Christopher Bradley
2007 IMO Shortlist G6 (USA)Determine the smallest positive real number k with the following property. Let $ABCD$ be a convex quadrilateral, and let points $A_1, B_1, C_1$ and $D_1$ lie on sides $AB, BC,CD$ and $DA$, respectively. Consider the areas of triangles $AA_1D_1, BB_1A_1, CC_1B_1$, and $DD_1C_1$; let $S $ be the sum of the two smallest ones, and let $S_1$ be the area of quadrilateral $A_1B_1C_1D_1$. Then we always have $ kS_1 \ge S $.
by Z. Feng & O. Golberg
2007 IMO Shortlist G7 (IRN)Given an acute triangle $ABC$ with angles $ \alpha, \beta $ and $ \gamma $ at vertices $A, B$ and $C$, respectively, such that $ \beta > \gamma $ . Point $I $ is the incenter, and $R$ is the circumradius. Point $D $ is the foot of the altitude from vertex $A$. Point $K$ lies on line $AD$ such that $AK = 2R$, and $D$ separates $A$ and $K$. Finally, lines $DI$ and $KI$ meet sides $AC$ and $BC$ at $E$ and $F$, respectively. Prove that if $IE = IF$ then $\beta \le 3 \gamma $.
by Davoud Vakili
2007 IMO Shortlist G8 (POL)Point $P$ lies on side $AB $ of a convex quadrilateral $ABCD$. Let $\omega$ be the incircle of triangle $CPD$, and let $I$ be its incenter. Suppose that $\omega$ is tangent to the incircles of triangles $APD$ and $BPC$ at points $K$ and $L$, respectively. Let lines $AC$ and $BD$ meet at $E$, and let lines $AK$ and $BL$ meet at $F$. Prove that points $E, I$, and $F$ are collinear.
by Waldemar Pompe
2008 IMO Shortlist G1 (RUS) problem 1In an acute-angled triangle $ABC$, point $H$ is the orthocentre and $A_o, B_o, C_o$ are the midpoints of the sides $BC, CA, AB$, respectively. Consider three circles passing through $H$: $\omega_a$ around $A_o, \omega_b$ around $B_o$ and $\omega_c$ around $C_o$. The circle $\omega_a$ intersects the line $BC$ at $A_1$ and $A_2$; $\omega_b$ intersects$ CA$ at $B_1$ and $B_2$; $\omega_c$ intersects $AB$ at $C_1$ and $C_2$. Show that the points $A_1, A_2, B_1, B_2, C_1, C_2$ lie on a circle.
by A. Gavrilyuk
2008 IMO Shortlist G2 (LUX)Given trapezoid $ABCD$ with parallel sides $AB$ and $CD$, assume that there exist points $ E$ on line $ BC$ outside segment $BC$, and F inside segment $AD$, such that $\angle DAE = \angle CBF$. Denote by $I$ the point of intersection of $CD$ and $EF$, and by $J$ the point of intersection of $AB$ and $EF$. Let $K$ be the midpoint of segment $EF$; assume it does not lie on line $AB$. Prove that $I$ belongs to the circumcircle of $ABK$ if and only if $K$ belongs to the circumcircle of $CDJ$.
by Charles Leytem
2008 IMO Shortlist G3 (PER)Let $ABCD$ be a convex quadrilateral and let $P$ and $ Q$ be points in $ABCD$ such that $ PQDA$ and $QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $E$ on the line segment $ PQ$ such that $ \angle PAE = \angle QDE$ and $\angle PBE =\angle QCE$. Show that the quadrilateral $ABCD$ is cyclic.
by John Cuya
2008 IMO Shortlist G4 (IRN)In an acute triangle $ABC$ segments$ BE $ and $CF $ are altitudes. Two circles passing through the points $A$ and $F$ are tangent to the line $BC$ at the points $P$ and $Q$ so that $B$ lies between $C$ and $Q$. Prove that the lines $PE$ and $QF$ intersect on the circumcircle of triangle $AEF$.
2008 IMO Shortlist G5 (NLD)
Let$ k$ and $ n $be integers with $ 0 \le k \le n - 2$ . Consider a set $L $ of $n $ lines in the plane such that no two of them are parallel and no three have a common point. Denote by $I $ the set of intersection points of lines in $L$. Let $O$ be a point in the plane not lying on any line of $L$. A point $X \in I $ is colored red if the open line segment $ OX $ intersects at most $k$ lines in $L$. Prove that $ I$ contains at least $ \frac{1}{2} (k+ 1)(k + 2) $red points.
2008 IMO Shortlist G6 (SRB)
There is given a convex quadrilateral $ABCD$. Prove that there exists a point $P$ inside the quadrilateral such that $ \angle PAB + \angle PDC = \angle PBC + \angle PAD = \angle PCD + \angle PBA = \angle PDA + \angle PCB = 90^o$ if and only if the diagonals $AC$ and $BD $ are perpendicular.
by Dušan Ðukic
2008 IMO Shortlist G7 (RUS) problem 6Let $ABCD$ be a convex quadrilateral with $AB \ne BC$. Denote by $\omega_1$ and $\omega_2$ the incircles of triangles $ABC$ and $ADC$. Suppose that there exists a circle $ \omega$ inscribed in angle $ABC$, tangent to the extensions of line segments $AD$ and $CD$. Prove that the common external tangents of $\omega_1 $ and $\omega_2 $ intersect on $ \omega$.
by V. Shmarov
2009 IMO Shortlist G1 (BEL) problem 4Let ABC be a triangle with $AB = AC$. The angle bisectors of $A$ and $B$ meet the sides $BC$ and $AC$ in $D$ and $E$, respectively. Let $K$ be the incenter of triangle $ADC$. Suppose that $\angle BEK = 45^o$. Find all possible values of $\angle BAC$.
by H. Lee, P. Vandendriessche & J. Vonk,
2009 IMO Shortlist G2 (RUS) problem 2Let $ABC$ be a triangle with circumcenter $O$. The points $P$ and $Q$ are interior points of the sides $CA$ and $AB$, respectively. The circle $k $ passes through the midpoints of the segments $BP, CQ$, and $PQ$. Prove that if the line $PQ $ is tangent to circle $ k$ then $ OP = OQ$.
by Sergei Berlov
2009 IMO Shortlist G3 (IRN)Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$ , respectively. Let $G $ be the point where the lines $BY $ and $CZ $ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCY R$ and $BCSZ$ are parallelograms. Prove that $GR = GS$.
2009 IMO Shortlist G4 (UNK)
Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and$ BD$ meet at $E $ and the lines $AD $ and $ BC $ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E, G,$ and $ H$.
2009 IMO Shortlist G5 (POL)
2009 IMO Shortlist G6 (UKR)
Let the sides $AD$ and $BC$ of the quadrilateral $ABCD$ (such that $AB $ is not parallel to $CD$) intersect at point $P$. Points $O_1 $ and $O_2 $ are the circumcenters and points $H_1 $ and $H_2$ are the orthocenters of triangles $ABP$ and $DCP$, respectively. Denote the midpoints of segments $O_1H_1$ and $O_2H_2$ by $E_1$ and $E_2$, respectively. Prove that the perpendicular from $E_1 $ on $ CD$, the perpendicular from $E_2$ on $AB$ and the line $H_1H_2$ are concurrent.
by Eugene Bilopitov
2009 IMO Shortlist G7 (IRN)Let $ABC $ be a triangle with incenter $ I$ and let $X, Y $ and $ Z$ be the incenters of the triangles $BIC, CIA$ and $AIB$, respectively. Let the triangle $ XY Z $ be equilateral. Prove that $ABC$ is equilateral too.
2009 IMO Shortlist G8 (BGR)
Let $ABCD$ be a circumscribed quadrilateral. Let $g$ be a line through $A$ which meets the segment $BC$ in $M$ and the line $CD $ in $N$. Denote by $ I_1, I_2$, and $ I_3$ the incenters of $\vartriangle ABM, \vartriangle MNC$, and $ \vartriangle NDA$, respectively. Show that the orthocenter of $\vartriangle I_1I_2I_3$ lies on $g$.
2010 IMO Shortlist G1 (UNK)
Let $ABC $ be an acute triangle with $D,E, F$ the feet of the altitudes lying on $BC,CA,AB$ respectively. One of the intersection points of the line $EF$ and the circumcircle is $P$. The lines $BP$ and $DF$ meet at point $Q$. Prove that $AP = AQ$.
by Christopher Bradley
2010 IMO Shortlist G2 (POL) problem 4Point $P$ lies inside triangle $ABC$. Lines $AP, BP, CP$ meet the circumcircle of $ABC$ again at points $K, L, M$, respectively. The tangent to the circumcircle at $C$ meets line $AB$ at $S$. Prove that $SC = SP$ if and only if $MK = ML$.
by Marcin E. Kuczma
2010 IMO Shortlist G3 (ARM)Let $A_1A_2...A_n $ be a convex polygon. Point $P$ inside this polygon is chosen so that its projections $P_1, ..., P_n$ onto lines $A_1A_2, ... , A_nA_1$ respectively lie on the sides of the polygon. Prove that for arbitrary points $X_1, ... , X_n$ on sides $A_1A_2, ... , A_nA_1$ respectively, $ max \{ \frac{X_1X_2}{P_1P_2},... , \frac{X_nX_1}{P_nP_1}\} \ge 1$.
by Nairi Sedrakyan
2010 IMO Shortlist G4 (HKG) problem 2Let $I$ be the incenter of a triangle $ABC$ and $ \Gamma $ be its circumcircle. Let the line $AI$ intersect $ \Gamma$ at a point $D \ne A$. Let $F$ and $E$ be points on side $BC$ and arc $BDC$ respectively such that $\angle BAF = \angle CAE < \frac{1}{2} \angle BAC$. Finally, let $G $ be the midpoint of the segment $IF$. Prove that the lines $DG$ and $EI$ intersect on $\Gamma$.
by Tai Wai Ming & Wang Chongli
2010 IMO Shortlist G5 (UKR)Let $ABCDE $ be a convex pentagon such that $BC// AE, AB = BC + AE$, and $ \angle ABC = \angle CDE$. Let $M$ be the midpoint of $CE$, and let $O $ be the circumcenter of triangle $BCD$. Given that $\angle DMO =90^o$ , prove that $2 \angle BDA = \angle CDE$.
by Nazar Serdyuk
2010 IMO Shortlist G6 (BUL)The vertices $X, Y , Z$ of an equilateral triangle $XYZ$ lie respectively on the sides $BC, CA, AB$ of an acute-angled triangle $ABC$. Prove that the incenter of triangle $ABC$ lies inside triangle $XYZ$.
by Nikolay Beluhov
Three circular arcs $\gamma_1, \gamma_2,$ and $\gamma_3$ connect the points $A$ and $C.$ These arcs lie in the same half-plane defined by line $AC$ in such a way that arc $\gamma_2$ lies between the arcs $\gamma_1$ and $\gamma_3.$ Point $B$ lies on the segment $AC.$ Let $h_1, h_2$, and $h_3$ be three rays starting at $B,$ lying in the same half-plane, $h_2$ being between $h_1$ and $h_3.$ For $i, j = 1, 2, 3,$ denote by $V_{ij}$ the point of intersection of $h_i$ and $\gamma_j$ (see the Figure below). Denote by $\widehat{V_{ij}V_{kj}}\widehat{V_{kl}V_{il}}$ the curved quadrilateral, whose sides are the segments $V_{ij}V_{il},$ $V_{kj}V_{kl}$ and arcs $V_{ij}V_{kj}$ and $V_{il}V_{kl}.$ We say that this quadrilateral is $circumscribed$ if there exists a circle touching these two segments and two arcs. Prove that if the curved quadrilaterals $\widehat{V_{11}V_{21}}\widehat{V_{22}V_{12}}, \widehat{V_{12}V_{22}}\widehat{V_{23}V_{13}},\widehat{V_{21}V_{31}}\widehat{V_{32}V_{22}}$ are circumscribed, then the curved quadrilateral $\widehat{V_{22}V_{32}}\widehat{V_{33}V_{23}}$ is circumscribed, too.
by Géza Kós
2011 IMO Shortlist G1 (EST)
Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose center $L$ lies on the side $BC$. Suppose that $ \omega$ is tangent to $AB$ at $B'$ and to $AC$ at $C'$ . Suppose also that the circumcenter $O$ of the triangle $ABC$ lies on the shorter arc $B' C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points.
by Härmel Nestra
2011 IMO Shortlist G2 (ISR)Let $A_1A_2A_3A_4 $ be a non-cyclic quadrilateral. Let $O_1 $ and $r_1 $ be the circumcenter and the circumradius of the triangle $A_2A_3A_4$. Define $O_2, O_3, O_4$ and $r_2, r_3, r_4$ in a similar way. Prove that $ \frac{1}{O_1A_1^2 - r_1^2}+ \frac{1}{O_2A_2^2 - r_2^2}+\frac{1}{O_3A_3^2 - r_3^2}+\frac{1}{O_4A_4^2 - r_4^2}= 0$.
by Alexey Gladkich
2011 IMO Shortlist G3 (BRA)Let A$BCD$ be a convex quadrilateral whose sides $AD$ and $BC$ are not parallel. Suppose that the circles with diameters $AB$ and $CD$ meet at points $E$ and $F$ inside the quadrilateral. Let $\omega_E$ be the circle through the feet of the perpendiculars from $E$ to the lines $AB, BC$, and $CD$. Let $ \omega_F$ be the circle through the feet of the perpendiculars from $F$ to the lines $CD, DA$, and $AB$. Prove that the midpoint of the segment $EF$ lies on the line through the two intersection points of $ \omega_E$ and $\omega_F$ .
by Carlos Shine
2011 IMO Shortlist G4 (RUS)Let $ABC$ be an acute triangle with circumcircle $\Omega$. Let $B_0$ be the midpoint of $AC$ and let $C_0$ be the midpoint of $AB$. Let $D$ be the foot of the altitude from $A$, and let $G$ be the centroid of the triangle $ABC$. Let $\omega $ be a circle through $B_0$ and $C_0$ that is tangent to the circle $\Omega$ at a point $X \ne A$. Prove that the points $D, G$, and $X$ are collinear.
by Ismail Isaev & Mikhail Isaev
2011 IMO Shortlist G5 (SVN)
Let $ABC$ be a triangle with incenter $I$ and circumcircle $\omega$. Let $D$ and $E$ be the second intersection points of $\omega$ with the lines $AI$ and $BI$, respectively. The chord $DE$ meets $AC$ at a point $F$, and $BC$ at a point $G$. Let $P$ be the intersection point of the line through $F$ parallel to $AD$ and the line through G parallel to $BE$. Suppose that the tangents to $\omega$ at $A$ and at $B$ meet at a point $K$. Prove that the three lines $AE, BD$, and $KP$ are either parallel or concurrent.
by Irena Majcen & Kris Stopar
2011 IMO Shortlist G6 (BEL)Let $ABC$ be a triangle with $AB = AC$, and let $D$ be the midpoint of $AC$. The angle bisector of $ \angle BAC$ intersects the circle through $D, B$, and $C$ in a point $E$ inside the triangle $ABC$. The line BD intersects the circle through $A, E$, and $B$ in two points $B$ and $F$. The lines $AF$ and $BE$ meet at a point $I$, and the lines $CI$ and $BD$ meet at a point $K$. Show that $I$ is the incenter of triangle $KAB$.
by Jan Vonk & Hojoo Lee
2011 IMO Shortlist G7 (JPN) Let $ABCDEF$ be a convex hexagon all of whose sides are tangent to a circle $\omega$ with center O. Suppose that the circumcircle of triangle $ACE$ is concentric with $\omega$ . Let $J$ be the foot of the perpendicular from $B$ to $CD$. Suppose that the perpendicular from $B$ to $DF$ intersects the line $EO$ at a point $K$. Let $L$ be the foot of the perpendicular from $K$ to $DE$. Prove that $DJ = DL$.
2011 IMO Shortlist G8 (JPN) problem 6
Let $ABC$ be an acute triangle with circumcircle $\omega$. Let $t$ be a tangent line to $ \omega.$ Let $t_a, t_b$, and $t_c$ be the lines obtained by reflecting $t$ in the lines $BC, CA$, and $AB$, respectively. Show that the circumcircle of the triangle determined by the lines $t_a, t_b$, and $t_c$ is tangent to the circle $\omega$.
2012 IMO Shortlist G1 (HEL) problem 1
In the triangle $ABC$ the point $J$ is the center of the excircle opposite to $A$. This excircle is tangent to the side $BC$ at $M$, and to the lines $AB$ and $AC$ at $K$ and $L$ respectively. The lines $LM$ and $BJ$ meet at $F$, and the lines $KM$ and $CJ$ meet at $G$. Let $S$ be the point of intersection of the lines $AF$ and $BC$, and let $T$ be the point of intersection of the lines $AG$ and $BC$. Prove that $M$ is the midpoint of $ST$.
by Evangelos Psychas
2012 IMO Shortlist G2Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ meet at $E$. The extensions of the sides $AD$ and $BC$ beyond $A$ and $B$ meet at $F$. Let $G$ be the point such that $ECGD$ is a parallelogram, and let $H$ be the image of $E$ under reflection in $AD$. Prove that $D, H, F, G$ are concyclic.
2012 IMO Shortlist G3
In an acute triangle $ABC$ the points $D, E$ and $F$ are the feet of the altitudes through $A, B$ and $C$ respectively. The incenters of the triangles $AEF$ and $BDF$ are $I_1$ and $I_2$ respectively; the circumcenters of the triangles $ACI_1$ and $BCI_2$ are $O_1 $ and $O_2$ respectively. Prove that $I_1I_2$ and $O_1O_2$ are parallel.
2012 IMO Shortlist G4
Let $ABC$ be a triangle with $AB \ne AC$ and circumcenter $O$. The bisector of $\angle BAC$ intersects $BC$ at $D$. Let $E$ be the reflection of $D$ with respect to the midpoint of $BC$. The lines through $D$ and $E$ perpendicular to $BC$ intersect the lines $AO$ and $AD$ at $X$ and $Y$ respectively. Prove that the quadrilateral $BXCY$ is cyclic.
2012 IMO Shortlist G5 (CZE) problem 5
Let $ABC$ be a triangle with $\angle BCA = 90^o$, and let $C_0$ be the foot of the altitude from $C$. Choose a point $X$ in the interior of the segment $CC_0$, and let $K, L$ be the points on the segments $AX,BX$ for which $BK = BC$ and $AL = AC$ respectively. Denote by $M$ the intersection of $AL$ and $BK$. Show that $MK = ML$.
by Josef Tkadlec
2012 IMO Shortlist G6Let $ABC$ be a triangle with circumcenter $O$ and incenter $I$. The points $D, E$ and $F$ on the sides $BC, CA$ and $AB$ respectively are such that $BD+ BF=CA$ and $CD + CE = AB$. The circumcircles of the triangles $BFD$ and $CDE$ intersect at $P \ne D$. Prove that $OP = OI$.
2012 IMO Shortlist G7
Let $ABCD$ be a convex quadrilateral with non-parallel sides $BC$ and $AD$. Assume that there is a point $E$ on the side $BC$ such that the quadrilaterals $ABED$ and $AECD$ are circumscribed. Prove that there is a point $F$ on the side $AD$ such that the quadrilaterals $ABCF$ and $BCDF$ are circumscribed if and only if $AB$ is parallel to $CD$.
2012 IMO Shortlist G8 (ROU)
Let $ABC$ be a triangle with circumcircle $\omega$ and $\ell$ a line without common points with $\omega$. Denote by $P$ the foot of the perpendicular from the center of $\omega$ to $\ell$. The side-lines $BC,CA,AB$ intersect $\ell$ at the points $X, Y, Z$ different from $P$. Prove that the circumcircles of the triangles $AXP,BYP$ and $CZP$ have a common point different from $P$ or are mutually tangent at $P$.
by Cosmin Pohoata
2013 IMO Shortlist G1 (THA) problem 4Let $ABC$ be an acute-angled triangle with orthocenter $H$, and let $W$ be a point on side $BC$. Denote by $M$ and $N$ the feet of the altitudes from $B$ and $C$, respectively. Denote by $\omega_1$ the circumcircle of $BWN$, and let $X$ be the point on $\omega_1$ which is diametrically opposite to $W$. Analogously, denote by $\omega_2$ the circumcircle of $CWM$, and let $Y$ be the point on $\omega_2$ which is diametrically opposite to $W$. Prove that $X, Y$ and $H$ are collinear.
by Warut Suksompong & Potcharapol Suteparuk
2013 IMO Shortlist G2 (IRN)Let $\omega$ be the circumcircle of a triangle $ABC$. Denote by $M$ and $N$ the midpoints of the sides $AB$ and $AC$, respectively, and denote by $T$ the midpoint of the arc $BC$ of $\omega$ not containing $A$. The circumcircles of the triangles $AMT$ and $ANT$ intersect the perpendicular bisectors of $AC$ and $AB$ at points $X$ and $Y$ , respectively; assume that $X$ and $Y$ lie inside the triangle $ABC$. The lines $MN$ and $XY$ intersect at $K$. Prove that $KA= KT$.
2013 IMO Shortlist G3 (SRB)
In a triangle $ABC$, let $D$ and $E$ be the feet of the angle bisectors of angles $A$ and $B$, respectively. A rhombus is inscribed into the quadrilateral $AEDB$ (all vertices of the rhombus lie on different sides of $AEDB$). Let $\phi$ be the non-obtuse angle of the rhombus. Prove that $\phi \le max\{\angle BAC,\angle ABC\}$.
2013 IMO Shortlist G4 (GEO)
Let $ABC$ be a triangle with $\angle B > \angle C$. Let $P$ and $Q$ be two different points on line $AC$ such that $\angle PBA =\angle QBA =\angle ACB$ and $A$ is located between $P$ and $C$. Suppose that there exists an interior point $D$ of segment $BQ$ for which $PD = PB$. Let the ray $AD$ intersect the circle $ABC$ at $R \ne A$. Prove that $QB = QR$.
2013 IMO Shortlist G5 (UKR)
Let $ABCDEF$ be a convex hexagon with $AB = DE, BC = EF, CD =FA$, and $\angle A - \angle D =\angle C - \angle F =\angle E- \angle B$. Prove that the diagonals $AD,BE$, and $CF$ are concurrent.
2013 IMO Shortlist G6 (RUS) problem 3
Let the excircle of the triangle $ABC$ lying opposite to $A$ touch its side $BC$ at the point $A_1$. Define the points $B_1$ and $C_1$ analogously. Suppose that the circumcentre of the triangle $A_1B_1C_1$ lies on the circumcircle of the triangle $ABC$. Prove that the triangle $ABC$ is right-angled.
by Alexander A. Polyansky
2014 IMO Shortlist G1 (GEO) problem 4The points $P$ and $Q$ are chosen on the side $BC$ of an acute-angled triangle $ABC$ so that $\angle PAB =\angle ACB$ and $\angle QAC =\angle CBA$. The points $M$ and $N$ are taken on the rays $AP$ and $AQ$, respectively, so that $AP = PM$ and $AQ =QN$. Prove that the lines $BM$ and $CN$ intersect on the circumcircle of the triangle $ABC$.
by Giorgi Arabidze
2014 IMO Shortlist G2 (EST)Let $ABC$ be a triangle. The points $K, L$, and $M$ lie on the segments $BC, CA$, and $AB$, respectively, such that the lines $AK, BL$, and $CM$ intersect in a common point. Prove that it is possible to choose two of the triangles $ALM, BMK$, and $CKL$ whose inradii sum up to at least the inradius of the triangle $ABC$.
2014 IMO Shortlist G3 (RUS)
Let $\Omega$ and $O$ be the circumcircle and the circumcentre of an acute-angled triangle $ABC$ with $AB > BC$. The angle bisector of $\angle ABC$ intersects $\Omega$ at $M \ne B$. Let $\Gamma$ be the circle with diameter $BM$. The angle bisectors of $\angle AOB$ and $\angle BOC$ intersect $\Gamma$ at points $P$ and $Q$, respectively. The point $R$ is chosen on the line $PQ$ so that $BR= MR$. Prove that $BR // AC$. (Here we always assume that an angle bisector is a ray.)
2014 IMO Shortlist G4 (UNK)
Consider a fixed circle $\Gamma$ with three fixed points $A, B$, and $C$ on it. Also, let us fix a real number $\lambda \in (0, 1)$. For a variable point $P \notin \{A,B,C\}$ on $\Gamma$, let $M$ be the point on the segment $CP $ such that $CM= \lambda \cdot CP$. Let $Q$ be the second point of intersection of the circumcircles of the triangles $AMP$ and $BMC$. Prove that as $P$ varies, the point $Q$ lies on a fixed circle.
2014 IMO Shortlist G5 (IRN) problem 3
Let $ABCD$ be a convex quadrilateral with $\angle B = \angle D =90^o$. Point H is the foot of the perpendicular from $A$ to $BD$. The points $S$ and $T$ are chosen on the sides $AB$ and $AD$, respectively, in such a way that $H$ lies inside triangle $SCT$ and $ \angle SHC - \angle BSC= 90^o , \angle THC - \angle DTC = 90^o$ . Prove that the circumcircle of triangle $SHT$ is tangent to the line $BD$.
2014 IMO Shortlist G6 (IRN)
Let $ABC$ be a fixed acute-angled triangle. Consider some points $E$ and $F$ lying on the sides $AC$ and $AB$, respectively, and let $M$ be the midpoint of $EF$. Let the perpendicular bisector of $EF$ intersect the line $BC$ at $K$, and let the perpendicular bisector of $MK$ intersect the lines $AC$ and $AB$ at $S$ and $T$, respectively. We call the pair $(E, F)$ interesting, if the quadrilateral $KSAT$ is cyclic. Suppose that the pairs $(E_1, F_1)$ and $(E_2, F_2)$ are interesting. Prove that $\frac{E_1E_2}{AB}= \frac{F_1F_2}{AC}$.
2014 IMO Shortlist G7 (USA)
Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I.$ Let the line passing through $I$ and perpendicular to $CI$ intersect the segment $BC$ and the arc $BC$ (not containing $A$) of $\Omega$ at points $U$ and $V$ , respectively. Let the line passing through $U$ and parallel to $AI $ intersect $AV$ at $X$, and let the line passing through $V$ and parallel to $AI$ intersect $AB$ at $Y$ . Let $W$ and $Z$ be the midpoints of $AX$ and $BC$, respectively. Prove that if the points $I, X$, and $Y$ are collinear, then the points $I, W$, and $Z$ are also collinear.
2015 IMO Shortlist G1
Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ =AH$.
2015 IMO Shortlist G2 (HEL) problem 4
Let $ABC$ be a triangle inscribed into a circle $\Omega$ with center $O$. A circle $\Gamma$ with center $A$ meets the side $BC$ at points $D$ and $E$ such that $D$ lies between $B$ and $E$. Moreover, let $F$ and $G$ be the common points of $\Gamma$ and $\Omega$. We assume that $F$ lies on the arc $AB$ of $\Omega$ not containing $C$, and $G$ lies on the arc $AC$ of $\Omega$ not containing $B$. The circumcircles of the triangles $BDF$ and $CEG$ meet the sides $AB$ and $AC$ again at $K$ and $L$, respectively. Suppose that the lines $FK$ and $GL$ are distinct and intersect at $X$. Prove that the points $A, X$, and $O$ are collinear.
by Evangelos Psychas & Silouanos Brazitikos
2015 IMO Shortlist G3Let $ABC$ be a triangle with $\angle C = 90^o$, and let $H$ be the foot of the altitude from $C.$ A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.
2015 IMO Shortlist G4
Let $ABC$ be an acute triangle, and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ again at $P$ and $Q$, respectively. Let $T$ be the point such that the quadrilateral $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of the triangle $ABC$. Determine all possible values of $BT / BM$.
2015 IMO Shortlist G5 (SLV)
Let $ABC$ be a triangle with $CA \ne CB$. Let $D, F$, and $G$ be the midpoints of the sides $AB, AC$, and $BC$, respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I' $ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$.
2015 IMO Shortlist G6 (UKR) problem 3
Let $ABC$ be an acute triangle with $AB >AC$, and let $\Gamma$ be its circumcircle. Let $H, M$, and $F$ be the orthocenter of the triangle, the midpoint of $BC$, and the foot of the altitude from $A$, respectively. Let $Q$ and $K$ be the two points on $\Gamma$ that satisfy $ \angle AQH= 90^o$ and $\angle QKH= 90^o$. Prove that the circumcircles of the triangles $KQH$ and $KFM$ are tangent to each other.
2015 IMO Shortlist G7 (BUL)
Let $ABCD$ be a convex quadrilateral, and let $P, Q, R$, and $S$ be points on the sides $AB, BC, CD$, and $DA$, respectively. Let the line segments $PR$ and $QS$ meet at $O$. Suppose that each of the quadrilaterals $APOS, BQOP, CROQ$, and $DSOR$ has an incircle. Prove that the lines $AC, PQ$, and $RS$ are either concurrent or parallel to each other.
by Nikolay Beluhov
2015 IMO Shortlist G8 (BUL)A triangulation of a convex polygon $\Pi$ is a partitioning of $\Pi$ into triangles by diagonals having no common points other than the vertices of the polygon. We say that a triangulation is a Thaiangulation if all triangles in it have the same area.
Prove that any two different Thaiangulations of a convex polygon $\Pi$ differ by exactly two triangles. (In other words, prove that it is possible to replace one pair of triangles in the first Thaiangulation with a different pair of triangles so as to obtain the second Thaiangulation.)
by Nikolay Beluhov
2016 IMO Shortlist G1 (BEL) problem 1
In a convex pentagon $ABCDE$, let $F$ be a point on $AC$ such that $\angle FBC = 90^o$. Suppose triangles ABF, ACD and ADE are similar isosceles triangles with $\angle FAB = \angle FBA = \angle DAC = \angle DCA = \angle EAD = \angle EDA$. Let M be the midpoint of $CF$. Point $X$ is chosen such that $AMXE$ is a parallelogram. Show that $BD,EM$ and $FX$ are concurrent.2016 IMO Shortlist G2 (TWN)
Let $ABC$ be a triangle with circumcircle $ \Gamma$ and incentre $I$. Let $M$ be the midpoint of side $BC$. Denote by $D$ the foot of perpendicular from $I$ to side $BC$. The line through $I$ perpendicular to $AI$ meets sides $AB$ and $AC$ at $F$ and $E$ respectively. Suppose the circumcircle of triangle $AEF$ intersects $ \Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $ \Gamma$ .
by Evan Chen
2016 IMO Shortlist G3 (IND)Let $B= (-1,0)$ and $C = (1, 0)$ be fixed points on the coordinate plane. A nonempty, bounded subset $S$ of the plane is said to be nice if
(i) there is a point $T$ in $S$ such that for every point $Q$ in $S$, the segment $TQ $ lies entirely in $S$; and
(ii) for any triangle $P_1P_2P_3$, there exists a unique point $A$ in $S$ and a permutation $\sigma$ of the indices $\{1, 2, 3\}$ for which triangles $ABC$ and $P_{\sigma (1)} P_{\sigma (2)}P_{\sigma (3)}$ are similar.
Prove that there exist two distinct nice subsets $S$ and $S' $ of the set $\{(x; y) : x \ge 0, y \ge 0 \}$ such that if $A \in S$ and $A' \in S' $ are the unique choices of points in (ii), then the product $BA \cdot BA'$ is a constant independent of the triangle $P_1P_2P_3$.
by C.R. Pranesachar
Let $ABC$ be a triangle with $AB = AC \ne B$C and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.
2016 IMO Shortlist G5
Let $D$ be the foot of perpendicular from $A$ to the Euler line (the line passing through the circumcentre and the orthocentre) of an acute scalene triangle $ABC$. A circle $\omega$ with centre $S$ passes through $A$ and $D$, and it intersects sides $AB$ and $AC$ at $X$ and $Y$ respectively. Let $P$ be the foot of altitude from $A$ to $BC$, and let $M$ be the midpoint of $BC$. Prove that the circumcentre of triangle $XSY$ is equidistant from $P$ and $M$.
2016 IMO Shortlist G6
Let $ABCD$ be a convex quadrilateral with $\angle ABC = \angle ADC < 90^o$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $E$ and $F$ respectively, and meet each other at point $P$. Let $M$ be the midpoint of AC and let $ \omega$ be the circumcircle of triangle BPD. Segments BM and DM intersect $ \omega$ again at $X$ and $Y$ respectively. Denote by $Q$ the intersection point of lines $XE$ and $Y F$. Prove that $PQ \perp AC$.
2016 IMO Shortlist G7
Let $I$ be the incentre of a non-equilateral triangle $ABC$, $I_A$ be the $A$-excentre, $I'_A$ be the reflection of $I_A$ in BC, and $\ell_A$ be the reflection of line $AI'_A$ in $AI$. Define points $I_B, I'_B$ and line $\ell_B$ analogously. Let $P$ be the intersection point of $\ell_A$ and $\ell_B$.
a) Prove that $P$ lies on line $OI$ where $O$ is the circumcentre of triangle $ABC$.
b) Let one of the tangents from $P$ to the incircle of triangle $ABC$ meet the circumcircle at points $X$ and $Y$ . Show that $\angle XIY = 120^o$.
2016 IMO Shortlist G8
Let $A_1,B_1$ and $C_1$ be points on sides $BC,CA$ and $AB$ of an acute triangle $ABC$ respectively, such that $AA_1,BB_1$ and $CC_1$ are the internal angle bisectors of triangle $ABC$. Let $I$ be the incentre of triangle $ABC$, and $H$ be the orthocentre of triangle $A_1B_1C_1$. Show that $AH + BH + CH \ge AI + BI+ CI$.
2017 IMO Shortlist G1 (ITA)
Let $ABCDE$ be a convex pentagon such that $AB = BC =CD$,$ \angle EAB = \angle BCD$, and $\angle EDC =\angle CBA$. Prove that the perpendiular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.
2017 IMO Shortlist G2 (LUX) problem 4
Let $R$ and $S$ be distint points on circle $\Omega$, and let $t$ denote the tangent line to $\Omega$ at $R$. Point $R' $ is the reflection of $R$ with respect to $S$. A point $I$ is chosen on the smaller arc $RS$ of $\Omega$ so that the circumcircle $\Gamma$ of triangle $ ISR' $ intersects $t$ at two different points. Denote by $A$ the common point of $ \Gamma$ and $t$ that is closest to $R$. Line $AI$ meets $\Omega$ again at $J$. Show that $JR' $ is tangent to $ \Gamma$.
by Charles Leytem
2017 IMO Shortlist G3 (UKR)Let $O$ be the circumcenter of an acute scalene triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.
2017 IMO Shortlist G4 (DEN)
In triangle $ABC$, let $\omega$ be the excircle opposite $A$. Let $D, E$, and $F$ be the points where \omega is tangent to lines $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.
2017 IMO Shortlist G5 (UKR)
Let $ABCC_1B_1A_1$ be a convex hexagon such that $AB = BC$, and suppose that the line segments $AA_1, BB_1$, and $CC_1$ have the same perpendicular bisector. Let the diagonals $AC_1$ and $A_1C$ meet at $D$, and denote by $\omega$ the circle $ABC$. Let $\omega$ intersect the circle $A_1BC_1$ again at $E \ne B$. Prove that the lines $BB_1 $ and $DE$ intersect on $\omega$.
2017 IMO Shortlist G6 (CZE)
Let $n \ge 3$ be an integer. Two regular $n$-gons $A$ and $B$ are given in the plane. Prove that the vertices of $A$ that lie inside $B$ or on its boundary are consecutive.
(That is, prove that there exists a line separating those vertices of $A$ that lie inside $B$ or on its boundary from the other vertices of $A$.)
2017 IMO Shortlist G7 (KZA)
A convex quadrilateral $ABCD$ has an inscribed circle with center $I$. Let $I_a, I_b, I_c$, and $I_d$ be the incenters of the triangles $DAB, ABC, BCD$, and $CDA$, respectively. Suppose that the common external tangents of the circles $AI_bI_d$ and $CI_bI_d$ meet at $X$, and the common external tangents of the circles $BI_aI_c$ and $DI_aI_c $ meet at $Y$ . Prove that $\angle XIY = 90^o$.
2017 IMO Shortlist G8 (AUS)
There are $2017$ mutually external circles drawn on a blackboard, such that no two are tangent and no three share a common tangent. A tangent segment is a line segment that is a vommon tangent to two circles, starting at one tangent point and ending at the other one. Luciano is drawing tangent segments on the blackboard, one at a time, so that no tangent segment intersects any other circles or previously drawn tangent segments. Luciano keeps drawing tangent segments until no more can be drawn. Find all possible numbers of tangent segments when he stops drawing.
2018 IMO Shortlist G1 (HEL) problem 1
Let $\Gamma$ be the circumcircle of acute triangle $ABC$. Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = AE$. The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ respectively. Prove that lines $DE$ and $FG$ are either parallel or they are the same line.
by Silouanos Brazitikos, Vangelis Psyxas & Michael Sarantis
2018 IMO Shortlist G2 (AUSTRALIA)
Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quarilateral $APXY$ is cyclic.
2018 IMO Shortlist G3 (SAF)
A circle $\omega$ with radius $1$ is given. A collection $T$ of triangles is called [i]good[/i], if the following conditions hold:
* each triangle from $T$ is inscribed in $\omega$;
* no two triangles from $T$ have a common interior point.
Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$.
A point $T$ is chosen inside a triangle $ABC$. Let $A_1$, $B_1$, $C_1$ be the reflections of $T$ in $BC$, $CA$, and $AB$, respectively. Let $\Omega$ be the circumcircle of $\triangle A_1B_1C_1$. The lines $A_1T$, $B_1T$, $C_1T$ meet $\Omega$ again at $A_2$, $B_2$, $C_2$, respectively. Prove that lines $AA_2$, $BB_2$, $CC_2$ meet on $\Omega$.
2018 IMO Shortlist G5 (DEN)
Let $ABC$ be a triangle with circumcircle $\omega$ and incentre $I$. A line $\ell$ intersects the lines $AI$, $BI$, and $CI$ at points $D$, $E$, and $F$, respectively, distinct from the points $A$, $B$, $C$, and $I$. The perpendicular bisectors $x$, $y$, and $z$ of the segments $AD$, $BE$, and $CF$, respectively determine a triangle $\Theta$. Show that the circumcircle of the triangle $\Theta$ is tangent to $\Omega$.
2018 IMO Shortlist G6 (POL) problem 6Let $ABC$ be a triangle with circumcircle $\omega$ and incentre $I$. A line $\ell$ intersects the lines $AI$, $BI$, and $CI$ at points $D$, $E$, and $F$, respectively, distinct from the points $A$, $B$, $C$, and $I$. The perpendicular bisectors $x$, $y$, and $z$ of the segments $AD$, $BE$, and $CF$, respectively determine a triangle $\Theta$. Show that the circumcircle of the triangle $\Theta$ is tangent to $\Omega$.
A convex quadrilateral $ABCD$ satisfies $AB\cdot CD = BC\cdot DA$. Point $X$ lies inside $ABCD$ so that $\angle{XAB} = \angle{XCD}\quad\,\,\text{and}\quad\,\,\angle{XBC} = \angle{XDA}.$ Prove that $\angle{BXA} + \angle{DXC} = 180^\circ$
2018 IMO Shortlist G7 (RUS)
Let $O$ be the circumcentre, and $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. Let $P$ be an arbitrary point on $\Omega$, distinct from $A$, $B$, $C$, and their antipodes in $\Omega$. Denote the circumcentres of the triangles $AOP$, $BOP$, and $COP$ by $O_A$, $O_B$, and $O_C$, respectively. The lines $\ell_A$, $\ell_B$, $\ell_C$ perpendicular to $BC$, $CA$, and $AB$ pass through $O_A$, $O_B$, and $O_C$, respectively. Prove that the circumcircle of triangle formed by $\ell_A$, $\ell_B$, and $\ell_C$ is tangent to the line $OP$.
2019 IMO Shortlist G1 (Nigeria)
Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.
2019 IMO Shortlist G2 (Vietnam)
Let $ABC$ be an acute-angled triangle and let $D, E$, and $F$ be the feet of altitudes from $A, B$, and $C$ to sides $BC, CA$, and $AB$, respectively. Denote by $\omega_B$ and $\omega_C$ the incircles of triangles $BDF$ and $CDE$, and let these circles be tangent to segments $DF$ and $DE$ at $M$ and $N$, respectively. Let line $MN$ meet circles $\omega_B$ and $\omega_C$ again at $P \ne M$ and $Q \ne N$, respectively. Prove that $MP = NQ$.
2019 IMO Shortlist G3 (UKR) problem 2
by Anton Trygub
2019 IMO Shortlist G4 (Australia)
Let $P$ be a point inside triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2$, and $C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$.
2019 IMO Shortlist G5 (Hungary)
Let $ABCDE$ be a convex pentagon with $CD= DE$ and $\angle EDC \ne 2 \cdot \angle ADB$.
Suppose that a point $P$ is located in the interior of the pentagon such that $AP =AE$ and $BP= BC$.
Prove that $P$ lies on the diagonal $CE$ if and only if area $(BCD)$ + area $(ADE)$ = area $(ABD)$ + area $(ABP)$.
2019 IMO Shortlist G6 (Slovakia)
Let $I$ be the incentre of acute-angled triangle $ABC$. Let the incircle meet $BC, CA$, and $AB$ at $D, E$, and $F,$ respectively. Let line $EF$ intersect the circumcircle of the triangle at $P$ and $Q$, such that $F$ lies between $E$ and $P$. Prove that $\angle DPA + \angle AQD =\angle QIP$.
2019 IMO Shortlist G7 (IND) problem 6
Let $I$ be the incentre of acute triangle $ABC$ with $AB\neq AC$. The incircle $\omega$ of $ABC$ is tangent to sides $BC, CA$, and $AB$ at $D, E,$ and $F$, respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ at $R$. Line $AR$ meets $\omega$ again at $P$. The circumcircles of triangle $PCE$ and $PBF$ meet again at $Q$. Prove that lines $DI$ and $PQ$ meet on the line through $A$ perpendicular to $AI$
by Anant Mudgal
2019 IMO Shortlist G8 (Australia)
Let $\mathcal L$ be the set of all lines in the plane and let $f$ be a function that assigns to each line $\ell\in\mathcal L$ a point $f(\ell)$ on $f(\ell)$. Suppose that for any point $X$, and for any three lines $\ell_1,\ell_2,\ell_3$ passing through $X$, the points $f(\ell_1),f(\ell_2),f(\ell_3)$, and $X$ lie on a circle. Prove that there is a unique point $P$ such that $f(\ell)=P$ for any line $\ell$ passing through $P$.
Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$. Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.
2020 IMO Shortlist G2 problem 1 (Poland)
Consider the convex quadrilateral $ABCD$. The point $P$ is in the interior of $ABCD$. The following ratio equalities hold:
\[\angle PAD:\angle PBA:\angle DPA=1:2:3=\angle CBP:\angle BAP:\angle BPC\]
Prove that the following three lines meet in a point: the internal bisectors of angles $\angle ADP$ and $\angle PCB$ and the perpendicular bisector of segment $AB$.
by Dominik Burek
2020 IMO Shortlist G3 (Slovakia)
Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other.
In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that\[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\] (A disk is assumed to contain its boundary.)
Let $ABCD$ be a cyclic quadrilateral. Points $K, L, M, N$ are chosen on $AB, BC, CD, DA$ such that $KLMN$ is a rhombus with $KL \parallel AC$ and $LM \parallel BD$. Let $\omega_A, \omega_B, \omega_C, \omega_D$ be the incircles of $\triangle ANK, \triangle BKL, \triangle CLM, \triangle DMN$.
Prove that the common internal tangents to $\omega_A$, and $\omega_C$ and the common internal tangents to $\omega_B$ and $\omega_D$ are concurrent.
Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other
Let $P$ be a point on the circumcircle of acute triangle $ABC$. Let $D,E,F$ be the reflections of $P$ in the $A$-midline, $B$-midline, and $C$-midline. Let $\omega$ be the circumcircle of the triangle formed by the perpendicular bisectors of $AD, BE, CF$. Show that the circumcircles of $\triangle ADP, \triangle BEP, \triangle CFP,$ and $\omega$ share a common point.
Let $ABC$ be a triangle with incenter $I$ and circumcircle $\Gamma$. Circles $\omega_{B}$ passing through $B$ and $\omega_{C}$ passing through $C$ are tangent at $I$. Let $\omega_{B}$ meet minor arc $AB$ of $\Gamma$ at $P$ and $AB$ at $M\neq B$, and let $\omega_{C}$ meet minor arc $AC$ of $\Gamma$ at $Q$ and $AC$ at $N\neq C$. Rays $PM$ and $QN$ meet at $X$. Let $Y$ be a point such that $YB$ is tangent to $\omega_{B}$ and $YC$ is tangent to $\omega_{C}$.
Show that $A,X,Y$ are collinear.
2020 IMO Shortlist G9 problem 6
Prove that there exists a positive constant $c$ such that the following statement is true:
Consider an integer $n > 1$, and a set $\mathcal S$ of $n$ points in the plane such that the distance between any two different points in $\mathcal S$ is at least 1. It follows that there is a line $\ell$ separating $\mathcal S$ such that the distance from any point of $\mathcal S$ to $\ell$ is at least $cn^{-1/3}$.
(A line $\ell$ separates a set of points S if some segment joining two points in $\mathcal S$ crosses $\ell$.)
Note. Weaker results with $cn^{-1/3}$ replaced by $cn^{-\alpha}$ may be awarded points depending on the value of the constant $\alpha > 1/3$.
Ting-Feng Lin and Hung-Hsun Hans Yu, Taiwan
2021 IMO Shortlist G7 problem 3
Let $D$ be an interior point of the acute triangle $ABC$ with $AB > AC$ so that $\angle DAB = \angle CAD.$ The point $E$ on the segment $AC$ satisfies $\angle ADE =\angle BCD,$ the point $F$ on the segment $AB$ satisfies $\angle FDA =\angle DBC,$ and the point $X$ on the line $AC$ satisfies $CX = BX.$ Let $O_1$ and $O_2$ be the circumcenters of the triangles $ADC$ and $EXD,$ respectively. Prove that the lines $BC, EF,$ and $O_1O_2$ are concurrent.
Mykhailo Shtandenko, Ukraine
2021 IMO Shortlist G2 problem 4
Let $\Gamma$ be a circle with centre $I$, and $A B C D$ a convex quadrilateral such that each of the segments $A B, B C, C D$ and $D A$ is tangent to $\Gamma$. Let $\Omega$ be the circumcircle of the triangle $A I C$. The extension of $B A$ beyond $A$ meets $\Omega$ at $X$, and the extension of $B C$ beyond $C$ meets $\Omega$ at $Z$. The extensions of $A D$ and $C D$ beyond $D$ meet $\Omega$ at $Y$ and $T$, respectively. Prove that\[A D+D T+T X+X A=C D+D Y+Y Z+Z C.\]
Dominik Burek, Poland and Tomasz Ciesla, Poland
The IMO Compendium A Collection of Problems Suggested for The International Mathematical-Olympiads 1959-2009, 2nd Edition
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