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IMO 1959-2019 112p

geometry problems from International Mathematical Olympiads (also known as IMO)
aops links are in the names

1959 - 2019

Construct a right triangle with given hypotenuse c such that the median drawn to the hypotenuse is the geometric mean of the two legs of the triangle.

An arbitrary point M is selected in the interior of the segment AB. The squares AMCD and MBEF are constructed on the same side of AB, with the segments AM and MB as their respective bases. The circles circumscribed about these squares, with centers P and Q, intersect at M and also at another point N. Let N΄ denote the point of intersection of the straight lines AF and BC.
a) Prove that the points N and N΄ coincide.
b) Prove that the straight lines MN pass through a fixed point S independent of the choice of M.
c) Find the locus of the midpoints of the segments PQ as M varies between A and B.

by Cezar Cosnita
Two planes, P and Q, intersect along the line p. The point A is given in the plane P, and the point C in the plane Q, neither of these points lies on the straight line p: Construct an isosceles trapezoid ABCD (with AB parallel to CD) in which a circle can be inscribed, and with vertices B and D lying in the planes P and Q respectively.

In a given right triangle ABC, the hypotenuse BC, of length a, is divided into n equal parts (n an odd integer). Let ω be the acute angle subtending,  from A, that segment which contains the midpoint of the hypotenuse. Let h be the length of the altitude to the hypotenuse of the triangle. Prove $\tan \omega =\frac{4nh}{\left( {{n}^{2}}-1 \right)\alpha }$                                                                                 

  by Gheorghe D. Simionescu

Construct triangle ABC, given h­­a , hb (the altitudes from A and B) and ma, the median from vertex A.

Consider the cube ABCDA'B'C'D' (with face ABCD directly above face A'B'C'D').
a) Find the locus of the midpoints of segments XY , where X is any point of AC and Y is any point of B'D'.
b) Find the locus of points Z which lie on the segments XY of part (a) such that ΖΥ = 2 ΧΖ .

An isosceles trapezoid with bases a and b and altitude h is given.
a) On the axis of symmetry of this trapezoid, find all points P such that both  legs of   the trapezoid subtend right angles at P.
b) Calculate the distance of P from either base.
c) Determine under what conditions such points P actually exist. (Discuss various cases that might arise).

Consider triangle P1P2P3 and a point P within the triangle. Lines P1P , P2P , P3P intersect the opposite sides in points Q1, Q2, Q3 respectively. Prove that, of the numbers $\frac{{{P}_{1}}P}{P{{Q}_{1}}},\frac{{{P}_{2}}P}{P{{Q}_{2}}},\frac{{{P}_{3}}P}{P{{Q}_{3}}}$  at least one is    2 and at least one is  ≥ 2.

Construct triangle ABC if AC = b, AB = c and <AMB = ω, where M is the midpoint of segment BC and ω < 90­o. Prove that a solution exists if and only if $b\varepsilon \varphi \frac{\omega }{2}\le c<b$ . In what case does the equality hold?

Consider a plane ε  and three non-collinear points A, B, C on the same side of ε, suppose the plane determined by these three points is not parallel to ε. In plane a take three arbitrary points A΄, B΄, C΄. Let L, M, N be the midpoints of segments AA΄, BB΄, CC΄΄, let G be the centroid of triangle LMN. (We will not consider positions of the points A΄, B΄, C΄such that the points L, M, N do not form a triangle.) What is the locus of point G as A΄, B΄, C΄. range independently over the plane ε ?

by Gheorghe D. Simionescu
Consider the cube ABCDA΄B΄C΄D΄ (ABCD and A΄B΄C΄D΄ are the upper and lower bases, respectively, and edges AA΄, BB΄, CC΄, DD΄ are parallel). The point X moves at constant speed along the perimeter of the square ABCD in the direction ABCDA, and the point Y moves at the same rate along the perimeter of the square B΄C΄CB in the direction B΄C΄CB΄ Β. Points X and Y begin their motion at the same instant from the starting positions A and B΄, respectively. Determine and draw the locus of the midpoints of the segments XY.

On the circle K there are given three distinct points A,B,C. Construct (using only straightedge and compasses) a fourth point D on K such that a circle can be inscribed in the quadrilateral thus obtained.

Consider an isosceles triangle. Let r be the radius of its circumscribed circle and ρ the radius of its inscribed circle. Prove that the distance d between the centers of these two circles is

The tetrahedron SABC has the following property: there exist five spheres, each tangent to the edges SA, SB, SC, BC, CA, AB or to their extensions.
(a) Prove that the tetrahedron SABC is regular.
(b) Prove conversely that for every regular tetrahedron five such spheres exist.

Point Α  and segment BC are given. Determine the locus of points in space which are vertices of right angles with one side passing through Α, and the other side intersecting the segment BC.

In an n-gon all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation α1  α2  ≥ ... ≥ αn . Prove that α1 =  α2  = ... = αn .

A circle is inscribed in triangle ABC with sides a, b, c, Tangents to the circle parallel to the sides of the triangle are constructed. Each of these tangents cuts off a triangle from ∆ ABC. In each of these triangles, a circle is inscribed. Find the sum of the areas of all four inscribed circles (in terms of a, b, c).

In tetrahedron ABCD, vertex D is connected with Do the centroid of ∆ ABC. Lines parallel to D Do are drawn through A, B and C: These lines intersect the planes BCD, CAD and ABD in points A­1, B­1 and C­1, respectively. Prove that the volume of ABCD is one third the volume of A­111o. Is the result true if point D­o is selected anywhere within ∆ ABC

Given the tetrahedron ABCD whose edges AB and CD have lengths a and b respectively. The distance between the skew lines AB and CD is d, and the angle between them is ω. Tetrahedron ABCD is divided into two solids by plane ε, parallel to lines AB and CD. The ratio of the distances of ε from AB and CD is equal to k. Compute the ratio of the volumes of the two solids obtained.

Consider ∆ OAB with acute angle AOB: Through a point M ≠ O perpendiculars are drawn to OA and OB, the feet of which are P and Q respectively. The point of intersection of the altitudes of ∆ OPQ is H. What is the locus of H if M is permitted to range over (a) the side AB, (b) the interior of ∆ OAB?

by Gheorghe D. Simionescu
Prove: The sum of the distances of the vertices of a regular tetrahedron from the center of its circumscribed sphere is less than the sum of the distances of these vertices from any other point in space.

In the interior of sides BC, CA, AB of triangle ABC, any points K, L, M  respectively, are selected. Prove that the area of at least one of the triangles AML, BKM, CLK is less than or equal to one quarter of the area of triangle ABC.

Let ABCD be a parallelogram with side lengths AB = α, AD = 1, and with <BAD = ω. If ∆ABD is acute, prove that the four circles of radius 1 with centers A,B,C,D cover the parallelogram if and only if  $a\le \cos \omega +\sqrt{3}\sin \omega$

Prove that if one and only one edge of a tetrahedron is greater than 1, then its volume is 1 / 8.

Let Ao BoCo and A1B1C1 be any two acute-angled triangles. Consider all triangles ABC that are similar to ∆ A1B1C1 (so that vertices A1,B1,C1 correspond to vertices A, B, C  respectively) and circumscribed about triangle AoBoCo  (where Ao  lies on BC, Bo  on CA, and Co on AB). Of all such possible triangles, determine the one with maximum area, and construct it.

by Tullio Viola
Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which are the sides of a triangle.

For each value of k =1,2,3,4,5  find necessary and sufficient conditions on the number a > 0 so that there exists a tetrahedron with k edges of length a, and the remaining 6-k edges of length 1.

A semicircular arc γ is drawn on AB as diameter. C is a point on γ other than A  and B, and D is the foot of the perpendicular from C to AB. We consider three circles, γ1, γ2, γ3, all tangent to the line ΑΒ. Of these, γ1 is inscribed in ∆ ABC, while γ2 and γ3 are both tangent to CD  and to γ, one on each side of CD . Prove that γ1, γ2 and γ3 have a second tangent in common.

Let M be a point on the side AB of ∆ABC. Let r1, r2 and r be the radii of the inscribed circles of triangles AMC,BMC and ABC. Let q1, q2  and q be the radii of the excircles of the same triangles that lie in the angle ACB. Prove that   $\frac{{{r}_{1}}}{{{q}_{1}}}\cdot \frac{{{r}_{2}}}{{{q}_{2}}}=\frac{r}{q}$

In the tetrahedron ABCD,  angle BDC is a right angle. Suppose that the foot H of the perpendicular from D to the plane ABC is the intersection of the altitudes of ∆ABC. Prove that (AB + BC + CA)2 6 (AD2 + BD2 + CD2). For what tetrahedra does equality hold?

Consider a convex polyhedron P1 with nine vertices A1A2, ... ,A9. Let Pi be the polyhedron obtained from P1 by a translation that moves vertex A1 to Ai (i = 2, 3, ... , 9): Prove that at least two of the polyhedra P1,P2, … , P9 have an interior point in common.

All the faces of tetrahedron ABCD are acute-angled triangles. We consider all closed polygonal paths of the form XYZTX defined as follows: X is a point on edge AB distinct from A and B; similarly, Y,Z, T are interior points of edges BC,CD,DA respectively. Prove:
a) If <DAB + <BCD ≠ <CDA + <ABC ,then among the polygonal paths, there is none of minimal length.
b) If <DAB + <BCD = <CDA + <ABC, then there are infinitely many shortest polygonal paths, their common length being 2AC sin(a/2),  where a = <BAC + <CAD + <DAB.

Prove that if n ≥ 4, every quadrilateral that can be inscribed in a circle can be dissected into n quadrilaterals each of which is inscribable in a circle.
Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.

Determine whether or not there exists a finite set M of points in space not lying in the same plane such that, for any two points A and B of M, one can select two other points C and D of M so that lines AB and CD are parallel and not coincident.

A soldier needs to check on the presence of mines in a region having the shape of an equilateral triangle. The radius of action of his detector is equal to half the altitude of the triangle. The soldier leaves from one vertex of the triangle. What path shouid he follow in order to travel the least possible distance and still accomplish his mission?

by Ðorde Dugošija
On the sides of an arbitrary triangle ABC  triangles ABR, BCP, CAQ are constructed externally with <CBP = <CAQ = 450   <BCP = <ACQ = 300, <ABR = <BAR = 15­­0 . Prove that  <QRP = 900 and QR = RP.

by Jan van de Craats
In a plane convex quadrilateral of area 32,  the sum of the lengths of two opposite sides and one diagonal is 16. Determine all possible lengths of the other diagonal.

Equilateral triangles ABK,BCL,CDM,DAN are constructed inside the square ABCD: Prove that the midpoints of the four segments KL,LM,MN,NK and the midpoints of the eight segments AK, BK,BL,CL,CM,DM,DN,AN are the twelve vertices of a regular dodecagon.

by Jan van de Craats
P is a given point inside a given sphere. Three mutually perpendicular rays from P intersect the sphere at points U, V, and W, Q denotes the  vertex diagonally opposite to P in the parallelepiped determined by PU, PV, and PW. Find the locus of Q for all such triads of rays from P.

by Murray Klamkin
In triangle ABC, AB = AC. A circle is tangent internally to the circumcircle of triangle ABC and also to sides AB, AC at P, Q, respectively.  Prove that the midpoint of segment PQ is the center of the incircle of triangle ABC.

by Murray Klamkin
Two circles in a plane intersect. Let A be one of the points of intersection. Starting simultaneously from A two points move with constant speeds, each point travelling along its own circle in the same sense. The two points return to A simultaneously after one revolution. Prove that there is a fixed point P in the plane such that, at any time, the distances from P to the moving points are equal.

by Nikolai Vasil'ev and Igor F. Sharygin
Given a plane π, a point P in this plane and a point Q not in π, find all points R in π such that the ratio $\frac{QP+PR}{QR}$  is a maximum.

by Murray Klamkin
P is a point inside a given triangle ABC. D,E, F are the feet of the perpendiculars from P to the lines BC, CA, AB respectively. Find all P for which $\frac{BC}{PD}+\frac{CA}{PE}+\frac{AB}{PF}$ is least.

by David Monk
Three congruent circles have a common point O and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incenter and the circumcenter of the triangle and the point O are collinear.

A non-isosceles triangle A1A2A3 is given with sides a1, a2, a3 (ai is the side opposite Ai). For all  i = 1, 2, 3, Mi is the midpoint of side ai, and Ti is the point where the incircle touches side ai. Denote by Si the reflection of Ti in the interior bisector of angle Ai. Prove that the lines M1S1, M2S2 and M3S3  are concurrent.

by Jan van de Craats
The diagonals AC and CE of the regular hexagon ABCDEF are divided by the inner points M and N, respectively, so that $\frac{AM}{AC}=\frac{CN}{CE}=r$. Determine r if B,M, and N are collinear.

by Jan van de Craats
Let A be one of the two distinct points of intersection of two unequal coplanar circles C1 and C2 with centers O1  and O2, respectively. One of the common tangents to the circles touches C1  at P1 and C2  at P2, while the other touches C1  at Q1  and C2  at Q2. Let M1 be the midpoint of P1Q1 and M2  be the midpoint of P2Q2. Prove that <O1AO2  = <M1AM2.

by Igor F. Sharygin
Let ABC be an equilateral triangle and E the set of all points contained in the three segments AB,BC and CA (including A, B and C). Determine whether, for every partition of E into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle. Justify your answer.

Let ABCD be a convex quadrilateral such that the line CD is a tangent to the circle on AB as diameter. Prove that the line AB is a tangent to the circle on CD as diameter if and only if the lines BC and AD are parallel.

by Laurentiu Panaitopol
A circle has center on the side AB of the cyclic quadrilateral ABCD. The other three sides are tangent to the circle. Prove that AD + BC = AB.

by Frank Budden
A circle with center O passes through the vertices A and C of triangle ABC and intersects the segments AB and BC again at distinct points K and N respectively. The circumscribed circles of the triangles ABC and EBN intersect at exactly two distinct points B and M. Prove that angle <OMB is a right angle.

by Igor F. Sharygin
A triangle A1A2A3 and a point P0 are given in the plane. We define As = As-3 for all s ≥ 4. We construct a set of points P1, P2, P3, . . . , such that Pk+1 is the image of Pk under a rotation with center Ak+1 through angle 120o  clockwise  (for k = 0, 1, 2, . . . ). Prove that if P1986 = P0, then the triangle A1A2A3  is equilateral.

 by Gengzhe Chang and Dongxu Qi
Let A, B be adjacent vertices of a regular n-gon (n ≥ 5) in the plane having center at O. A triangle XY Z, which is congruent to and initially coincides with OAB, moves in the plane in such a way that Y and Z each trace out the whole boundary of the polygon, X remaining inside the polygon. Find the locus of X.

by Sven Sigurðsson
In an acute-angled triangle ABC the interior bisector of the angle A intersects BC at L and intersects the circumcircle of ABC again at N. From point L perpendiculars are drawn to AB and AC, the feet of these perpendiculars being K and M respectively. Prove that the quadrilateral AKNM and the triangle ABC have equal areas.

by I.A. Kushnir
Consider two concentric circles of radii R and r (R > r) with the same center. Let P be a fixed point on the smaller circle and B a variable point on the larger circle. The line BP meets the larger circle again at C. The perpendicular l to BP at P meets the smaller circle again at A. (If l is tangent to the circle at P then A P.)
(i) Find the set of values of BC­­2 + CA­­2 + AB­­2.
(ii) Find the locus of the midpoint of BC.

by Lucien Kieffer
ABC is a triangle right-angled at A, and D is the foot of the altitude from A. The straight line joining the incenters of the triangles ABD, ACD intersects the sides AB, AC at the points K, L respectively. S and T denote the areas of the triangles ABC and AKL respectively. Show that    S ≥ 2T.

by Dimitris Kontogiannis
In an acute-angled triangle ABC the internal bisector of angle A meets the circumcircle of the triangle again at A1. Points B1 and C1  are defined similarly. Let A0 be the point of intersection of the line AA1 with the external bisectors of angles B and C. Points B0 and C0 are defined similarly. Prove that:
(i) The area of the triangle A0B0C0 is twice the area of the hexagon AC1BA1CB1.
(ii) The area of the triangle A0B0C0 is at least four times the area of the triangle ABC.

by Esther Szekeres
Let ABCD be a convex quadrilateral such that the sides AB, AD, BC satisfy AB = AD + BC. There exists a point P inside the quadrilateral at a distance h from the line CD such that AP = h + AD and BP = h + BC. Show that:  $\frac{1}{\sqrt{h}}\ge \frac{1}{\sqrt{AD}}+\frac{1}{\sqrt{BC}}$.

by Eggert Briem
Chords AB and CD of a circle intersect at a point E inside the circle. Let M be an interior point of the segment EB. The tangent line at E to the circle through D, E, and M intersects the lines BC and AC at F and G, respectively. If $\frac{AM}{AB}=t$ , find  $\frac{EF}{EG}$ in terms of t.

by C.R. Pranesachar
Given a triangle ABC, let I be the center of its inscribed circle. The internal bisectors of the angles A,B,C  meet the opposite sides in A’, B’ , C’ respectively. Prove that $\frac{1}{4}<\frac{AI\cdot BI\cdot CI}{AA'\cdot BB'\cdot CC'}\le \frac{8}{27}$

by Arkadii Skopenkov
Let ABC be a triangle and P an interior point of ABC . Show that at least one of the angles <PAB, <PBC, <PCA is less than or equal  to 300.

by Johan Yebbou
In the plane let C be a circle, L a line tangent to the circle C  and M a point on L. Find the locus of all points P with the following  property: there exists two points Q,R on L such that M is the midpoint of QR and C is the inscribed circle of triangle PQR.

by Johan Yebbou
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 Donco Dimovski
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 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 Hans Lausch & G. Tonoyan
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 G5 (NZL) problem 5
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 $AG+GB+GH+DH +HE \ge CF$.

by Alastair McNaughton
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 Jeffrey P. Grossman
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 Nairi M. Sedrakyan
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
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 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 Vyacheslav Yasinskiy
1999 IMO Shortlist G3 (EST) problem 1
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.

by Jan Villemson
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 Pavel Kozhevnikov
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 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 Lev Emelyanov & Tatiana Emelyanova

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  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 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 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 Vyacheslav Yasinskiy
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 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
2004 IMO Shortlist G1 (ROU) problem 1
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 Dinu Serbanescu & Valentin Vornicu
2004 IMO Shortlist G4 (POL) problem 5
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 $AP = CP$ .

by Waldemar Pompe
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 G4 (POL) problem 5
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
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 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 4
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.

by Marek Pechal
2007 IMO Shortlist G4 (LUX) problem 2
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 $DAB$.

by Charles Leytem
2008 IMO Shortlist G1 (RUS) problem 1
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.

byAndrey Gavrilyuk 
2008 IMO Shortlist G7 (RUS) problem 6
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$.

by Vladimir Shmarov
2009 IMO Shortlist G1 (BEL) problem 4
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   $\angle BAC$.

by H. Lee, P. Vandendriessche & J. Vonk,
2009 IMO Shortlist G2 (RUS) problem 2
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$.

by Sergei Berlov
2010 IMO Shortlist G2 (POL) problem 4
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 $MK = ML$.

by Marcin E. Kuczma
2010 IMO Shortlist G4 (HKG) problem 2
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 $\Gamma$.

by Tai Wai Ming & Wang Chongli
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 $\omega$.

by Japanese PSC

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 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
2013 IMO Shortlist G1 (THA) problem 4
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.

by Warut Suksompong & Potcharapol Suteparuk
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 4
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 $ABC$.

by Giorgi Arabidze
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 $BD$.

by Ali Zamani
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 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.

by Danylo Khilko & Mykhailo Plotnikov
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.

by Art Waeterschoot
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
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 G6  (POL) problem 6
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 $\angle{BXA} + \angle{DXC} = 180^\circ$

by Tomasz Ciesla

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.
by Anton Trygub

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

source for shortlists:
The IMO Compendium A Collection of Problems Suggested for The International Mathematical-Olympiads 1959-2009, 2nd Edition

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