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geo shortlists

geometry problems from geometry Shortlists  with aops links in the names

besides
IMO , Balkan MO (BMO), JBMO,


complete at the moment/ also posted at:
Malaysia 2015 (inside aops here)
Pan African 2018 (inside aops here)
Thailand MO 2008-12, 2014  (inside aops here)


(in chronological order)



2008 Indonesia INAMO Geometry Shortlist

Given triangle $ ABC$. Points $ D,E,F$ outside triangle $ ABC$ are chosen such that triangles $ ABD$, $ BCE$, and $ CAF$ are equilateral triangles. Prove that cicumcircles of these three triangles are concurrent.

Let $ABC$ be an isosceles triangle right at $C$ and $P$ any point on $CB$. Let also $Q$ be the midpoint of $AB$ and $R, S$ be the points on $AP$ such that $CR$ is perpendicular to $AP$ and $|AS|=|CR|$. Prove that the $|RS|  =  \sqrt2 |SQ|$.

Given triangle $ABC$. A circle $\Gamma$ is tangent to the circumcircle of triangle $ABC$ at $A$ and tangent to $BC$ at $D$. Let $E$ be the intersection of circle $\Gamma$ and $AC$. Prove that
$$R^2=OE^2+CD^2\left(1- \frac{BC^2}{AB^2+AC^2}\right)$$where $O$ is the center of the circumcircle of triangle $ABC$, with radius $R$.

Given that two circles $\sigma_1$ and $\sigma_2$ internally tangent at $N$ so that $\sigma_2$ is inside $\sigma_1$. The points $Q$ and $R$ lies at $\sigma_1$ and $\sigma_2$, respectively, such that $N,R,Q$ are collinear. A line through $Q$ intersects $\sigma_2$ at $S$ and intersects $\sigma_1$ at $O$. The line through $N$ and $S$ intersects $\sigma_1$ at $P$. Prove that$$\frac{PQ^3}{PN^2} = \frac{PS \cdot RS}{NS}.$$

Let $ABCD$ be quadrilateral inscribed in a circle. Let $M$ be the midpoint of the segment $BD$. If the tangents of the circle at $ B$, and at $D$ are also concurrent with the extension of $AC$, prove that $\angle AMD = \angle CMD$.

Given triangle $ ABC$ with sidelengths $ a,b,c$. Tangents to incircle of $ ABC$ that parallel with triangle's sides form three small triangle (each small triangle has 1 vertex of $ ABC$). Prove that the sum of area of incircles of these three small triangles and the area of incircle of triangle $ ABC$ is equal to
$ \frac{\pi (a^{2}+b^{2}+c^{2})(b+c-a)(c+a-b)(a+b-c)}{(a+b+c)^{3}}$

Given an isosceles trapezoid $ABCD$ with base $AB$. The diagonals $AC$ and $BD$ intersect at point $S$. Let $M$ the midpoint of $BC$ and the bisector of the angle $BSC$ intersect $BC$ at $N$. Prove that $\angle AMD = \angle AND$.

Prove that there is only one triangle whose sides are consecutive natural numbers and one of the angles is twice the other angle.

Given a triangle $ABC$, the points $D$, $E$, and $F$ lie on the sides $BC$, $CA$, and $AB$, respectively, are such that
$$DC + CE = EA + AF = FB + BD.$$Prove that$$DE + EF + FD \ge \frac12 (AB + BC + CA).$$

Given a triangle $ABC$ with $AB = AC$, angle $\angle A = 100^o$ and $BD$ bisector of angle $\angle B$. Prove that$$BC = BD + DA.$$


2008 Thailand Geometry Shortlist


Point $O$ is the center of the circle with chords $AB$, $CD$, $EF$ with lengths $2, 3, 4$ units, respectively, viewed from the angle at point $O$ at an angle $\alpha$, $\beta$, $\alpha + \beta$ respectively. If $\alpha + \beta$ is less than $180$ degrees, find the length of the radius of the circle.

Let $\vartriangle ABC$ be a triangle with $\angle BAC = 90^o$ and $\angle ABC = 60^o$. Point $E$ is chosen on side $BC$ so that $BE : EC = 3 : 2$. Compute $\cos\angle CAE$.

Let $ABC$ be a triangle, let $D$ and $E$ be points on sides $AC$ and $BC$ respectively, such that $DE \parallel AB$ . Segments $AE$ and $BD$ intersect at $P$. The area of $\vartriangle ABP$ is $36$ square units and the area of $\vartriangle DEP$ is $25$ square units. Find the area of $\vartriangle ABC$.

Let $AD$ be the common chord of two equal-sized circles $O_1$ and $O_2$. Let $B$ and $C$ be points on $O_1$ and $O_2$, respectively, so that $D$ lies on the segment $BC$. Assume that $AB = 15, AD = 13$ and $BC = 18$, what is the ratio between the inradii of $\vartriangle ABD$ and $\vartriangle ACD$?

Let $ABC$ be a triangle. Angle bisectors of $\angle B$ and $\angle C$ intersect at point $O$ and intersect sides $AC$ and $AB$ at points $D$ and $E$ respectively If $OD=OE$, prove that$ \angle ABC=\angle ACB$ or $\angle ABC+\angle  ACB=120^o$.

$ABCD$ is quadrilateral with $AB=2$ , $BC=3$, $CD=7$ and $AD=6$ and $\angle ABC=90^o$. Prove that $ABCD$ is tangential and find the radius of the inscribed circle.

Triangle ABC is a triangle inscribed in a circle . Chord $CD$ bisects angle $\angle ACB$ , cut side $AB$ at point $X$, cuts the circumscribed circle at point $D$. Prove that $\frac{CX}{CA}+\frac{CX}{CB}=\frac{BA}{BD}$.

The segments $AC=6$ and $BD =4$ intersect at point $O$ at a right angle, such that $AO=2$ and $OD=3$. Lines $AB$ and $DC$ intersect at $E$ , line $EO$ intersects segment $AD$ at point $F$. Find the length of $EF$.

Let $ABC$ to be a triangle with $BC=2551$, $AC=2550$ and $AB=2549$with AD the altitude . Let the inscribed circle of triangle $BAD$ intersect $AD$ at point $E$. Let the inscribed circle of triangle $CAD$ intersect $AD$ at point $F$. Find the length of $EF$.

Let $P$ be a point outside a circle $\omega$. The tangents from $P$ to $\omega$ are drawn touching $\omega$ at points $A$ and $B$. Let $M$ and $N$ be the midpoints of $AP$ and $AB$, respectively. Line $MN$ is extended to cut $\omega$ at $C$ so that $N$ lies between $M$ and $C$. Line $PC$ intersects $\omega$ again at $D$, and lines $ND$ and $PB$ intersect at $O$. Prove that $MNOP$ is a rhombus.

2009 Thailand Geometry Shortlist


Let $ABC$ be a triangle with the median $CD$. Point $E$ lies on side $BC$ such that $EC=\frac13 BC$ and $\angle ABC=20^o$. $AE$ intersects $CD$ at point $O$ and $\angle DAO = \angle ADO$. Find the measure of the angle $\angle ACB$.

In triangle $\vartriangle ABC$, $D$ is the midpoint of $BC$. Points $E$ and $F$ are chosen on side $AC$ so that $AF = F E = EC$. Let $AD$ intersect $BE$ and $BF$ and $G$ and $H$, respectively. Find the ratio of the areas of $\vartriangle BGH$ and $\vartriangle ABC$.

Let $ABCD$ be a square with a side length of $ 1$ and $O$ be the midpoint of $AD$. A semicircle having $AD$ as diameter is drawn inside the square. Let $E$ be on the side $AB$ such that $CE$ is tangent to the circle of center $O$. Find the area of the triangle $CBE$.

Let $ABCD$ be a convex quadrilateral with side lengths $AB=BC=2$, $CD=2\sqrt3$, $DA=2\sqrt5$. Let $M$ and $N$ be the midpoints of diagonal $AC$ and $BD$ respectively and $MN=\sqrt2$. Find the area of the quadrilateral $ABCD$.

Rectangle $HOMF$ has $HO=11$ and $OM=5$ .Triangle $\vartriangle ABC$ has orthocenter $H$ and circumcenter $O$. The midpoint of side $BC$ is $M$ and the point that the altitude from $A$ meets $BC$ is $F$ . Find the length of $BC$.

In triangle $\vartriangle ABC, D$ and $E$ are midpoints of the sides $BC$ and $AC$, respectively. Lines $AD$ and $BE$ are drawn intersecting at $P$. It turns out that $\angle CAD = 15^o$ and $\angle APB = 60^o$. What is the value of $AB/BC$ ?

Let $\vartriangle ABC$ be a triangle with $AB > AC$, its incircle is tangent to $BC$ at $D$. Let $DE$ be a diameter of the incircle, and let $F$ be the intersection between line $AE$ and side $BC$. Find the ratio between the areas of $\vartriangle DEF$ and $\vartriangle ABC$ in terms of the three side lengths of$\vartriangle ABC$.

Let $O$ be the center of the circumcircle of the acute triangle $ABC$. $AO$ intersects $BC$ at point $D$. Let $S$ be the point on $BO$ such that $DS \parallel AB$. AS intersects $BC$ at point $T$. Prove that points $D,O,S,T$ lie on the same circle if and only if triangle $ABC$ is an isosceles triangle with $A$ as the vertex.

Let $ABCD$ be a convex quadrilateral with the property that $MA \cdot  MC + MA  \cdot  CD = MB  \cdot  MD$, where $M$ is the intersection of the diagonals $AC$ and $BD$. The angle bisector of $\angle ACD$ is drawn intersecting ray $\overrightarrow{BA}$ at $K$. Prove that $BC = DK$ if and only if $AB \parallel CD$.

Let $M$ be a point on the side $AC$ of the acute triangle $ABC$ .Let $N$ be a point on the extension of $AC$ beyond $C$ such that causes $MN=AC$. Let $D$ and $E$ be the projections of $M$ and $N$ on the lines $BC$ and $AB$, respectively. Prove that the orthocenter of triangle $ABC$, lies on the circumcircle of the triangle $BED$.

2009 Indonesia INAMO Geometry Shortlist


Given triangle $ABC$, $AL$ bisects angle $\angle BAC$ with $L$ on side $BC$. Lines $LR$ and $LS$ are parallel to $BA$ and $CA$ respectively, $R$ on side $AC$ and$ S$ on side $AB$, respectively. Through point $B$ draw a perpendicular on $AL$, intersecting $LR$ at $M$. If point $D$ is the midpoint of $BC$, prove that that the three points $A, M, D$ lie on a straight line.

For every triangle $ ABC$, let $ D,E,F$ be a point located on segment $ BC,CA,AB$, respectively. Let $ P$ be the intersection of $ AD$ and $ EF$. Prove that:
\[ \frac{AB}{AF}\times DC+\frac{AC}{AE}\times DB=\frac{AD}{AP}\times BC\]

Given a quadrilateral $ABCD$ inscribed in circle $\Gamma$.From a point P outside $\Gamma$, draw tangents $PA$ and $PB$ with $A$ and $B$ as touspoints. The line $PC$ intersects $\Gamma$ at point $D$. Draw a line through $B$ parallel to $PA$, this line intersects $AC$ and $AD$ at points $E$ and $F$ respectively. Prove that $BE = BF$.

Let $D, E, F$, be the touchpoints of the incircle in triangle $ABC$ with sides $BC, CA, AB$, respectively, . Also, let $AD$ and $EF$ intersect at $P$. Prove that$$\frac{AP}{AD} \ge 1 - \frac{BC}{AB + CA}$$.

Two circles intersect at points $A$ and $B$. The line $\ell$ through A intersects the circles at $C$ and $D$, respectively. Let $M, N$ be the midpoints of arc $BC$ and arc $BD$. which does not contain $A$, and suppose that $K$ is the midpoint of the segment $CD$ . Prove that $\angle MKN=90^o$.

Suppose the points $D, E, F$ lie on sides $BC, CA, AB$, respectively, so that $AD, BE, CF$ are the altitudes. Also, let $AD$ and $EF$ intersect at $P$. Prove that$$\frac{AP}{AD} \ge 1 - \frac{BC^2}{AB^2 + CA^2}$$

Given a convex quadrilateral $ABCD$, such that $OA = \frac{OB \cdot OD}{OC + CD}$ where $O$ is the intersection of the two diagonals. The circumcircle of triangle $ABC$ intersects $BD$ at point $Q$. Prove that $CQ$ bisects $\angle ACD$

Suppose the points $D, E, F$ lie on sides $BC, CA, AB$, respectively, so that $AD, BE, CF$ are angle bisectors. Define $P_1$, $P_2$, $P_3$ respectively as the intersection point of $AD$ with $EF$, $BE$ with $DF$, $CF$ with $DE$ respectively. Prove that
$$\frac{AD}{AP_1}+\frac{BE}{BP_2}+\frac{CF}{CP_3} \ge 6$$

Given triangle $ABC$. Let $A_1B_1$, $A_2B_2$,$ ...$, $A_{2008}B_{2008}$ be $2008$ lines parallel to $AB$ which divide triangle $ABC$ into $2009$ equal areas. Calculate the value of$$\big \lfloor \frac{A_1B_1}{2A_2B_2} + \frac{A_1B_1}{2A_3B_3} + ... + \frac{A_1B_1}{2A_{2008}B_{2008}} \rfloor$$

Given a triangle $ABC$ with incenter $I$ . It is known that $E_A$ is center of the ex-circle tangent to $BC$. Likewise, $E_B$ and $E_C$ are the centers of the ex-circles tangent to $AC$ and $AB$, respectively. Prove that $I$ is the orthocenter of the triangle $E_AE_BE_C$.

Given an acute triangle $ ABC$. The incircle of triangle $ ABC$ touches $ BC,CA,AB$ respectively at $ D,E,F$. The angle bisector of $ \angle A$ cuts $ DE$ and $ DF$ respectively at $ K$ and $ L$. Suppose $ AA_1$ is one of the altitudes of triangle $ ABC$, and $ M$ be the midpoint of $ BC$.
(a) Prove that $ BK$ and $ CL$ are perpendicular with the angle bisector of $ \angle BAC$.
(b) Show that $ A_1KML$ is a cyclic quadrilateral.

In triangle $ABC$, the incircle is tangent to $BC$ at $D$, to $AC$ at $E$, and to $AB$ at $F$. Prove that:
$$\frac{CE-EA}{\sqrt{AB}}+\frac{AF-FB}{\sqrt{BC}} +\frac{BD-DC}{\sqrt{CA}} \ge \frac{BD-DC}{\sqrt{AB}}
+\frac{CE-EA}{\sqrt{BC}} +\frac{AF-FB}{\sqrt{CA}}$$

2010 Indonesia INAMO Geometry Shortlist


In triangle $ABC$, let $D$ be the midpoint of $BC$, and $BE$, $CF$ are the altitudes. Prove that $DE$ and $DF$ are both tangents to the circumcircle of triangle $AEF$

Given an acute triangle $ABC$. The inscribed circle of triangle $ABC$ is tangent to $AB$ and $AC$ at $X$ and $Y$ respectively. Let $CH$ be the altitude. The perpendicular bisector of the segment $CH$ intersects the line $XY$ at $Z$. Prove that $\angle BZC = 90^o.$

Suppose $L_1$ is a circle with center $O$, and $L_2$ is a circle with center $O'$. The circles intersect at $ A$ and $ B$ such that $\angle OAO' = 90^o$. Suppose that point $X$ lies on the circumcircle of triangle $OAB$, but lies inside $L_2$. Let the extension of $OX$ intersect $L_1$ at $Y$ and $Z$. Let the extension of $O'X$ intersect $L_2$ at $W$ and $V$ . Prove that $\vartriangle XWZ$ is congruent with $\vartriangle  XYV$.

Given an acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$. Let $K$ be a point inside $ABC$ which is not $O$ nor $H$. Point $L$ and $M$ are located outside the triangle $ABC$ such that $AKCL$ and $AKBM$ are parallelogram. At last, let $BL$ and $CM$ intersects at $N$, and let $J$ be the midpoint of $HK$. Show that $KONJ$ is also a parallelogram.

Raja Oktovin, Pekanbaru
Given an arbitrary triangle $ABC$, with $\angle A = 60^o$ and $AC < AB$. A circle with diameter $BC$, intersects $AB$ and $AC$ at $F$ and $E$, respectively. Lines $BE$ and $CF$ intersect at $D$. Let $\Gamma$ be the circumcircle of $BCD$, where the center of $\Gamma$ is $O$. Circle $\Gamma$ intersects the line $AB$ and the extension of $AC$ at $M$ and $N$, respectively. $MN$ intersects $BC$ at $P$. Prove that points $A$, $P$, $O$ lie on the same line.

Given an acute triangle $ABC$ with $AC>BC$ and the circumcenter of triangle $ABC$ is $O$. The altitude of triangle $ABC$ from $C$ intersects $AB$ and the circumcircle at $D$ and $E$, respectively. A line which passed through $O$ which is parallel to $AB$ intersects $AC$ at $F$. Show that the line $CO$, the line which passed through $F$ and perpendicular to $AC$, and the line which passed through $E$ and parallel with $DO$ are concurrent.

Fajar Yuliawan, Bandung
In triangle $ABC$, find the smallest possible value of$$|(\cot A + \cot B)(\cot B +\cot C)(\cot C + \cot A)|$$

Given an acute triangle $ABC$ and points $D$, $E$, $F$ on sides $BC$, $CA$ and $AB$, respectively. If the lines $DA$, $EB$ and $FC$ are the angle bisectors of triangle $DEF$, prove that the three lines are the altitudes of triangle $ABC$.

Given two circles $\Gamma_1$ and $\Gamma_2$ which intersect at points $A$ and $B$. A line through $A$ intersects $\Gamma_1$ and $\Gamma_2$ at points $C$ and $D$, respectively. Let $M$ be the midpoint of arc $BC$ in $\Gamma_1$ ,which does not contains $A$, and $N$ is the midpoint of the arc $BD$ in $\Gamma_2$, which does not contain $A$. If $K$ is the midpoint of $CD$, prove that $\angle MKN = 90^o.$

Given two circles with one of the centers of the circle is on the other circle. The two circles intersect at two points $C$ and $D$. The line through $D$ intersects the two circles again at $A$ and $ B$. Let $H$ be the midpoint of the arc $AC$ that does not contain $D$ and the segment $HD$ intersects circle that does not contain $H$ at point $E$. Show that $E$ is the center of the incircle of the triangle $ACD$.

Given triangle $ABC$ and point $P$ on the circumcircle of triangle $ABC$. Suppose the line $CP$ intersects line $AB$ at point $E$ and line $BP$ intersect line $AC$ at point $F$. Suppose also the perpendicular bisector of $AB$ intersects $AC$ at point $K$ and the perpendicular bisector of $AC$ intersects $AB$ at point $J$. Prove that$$\left( \frac{CE}{BF}\right)^2= \frac{AJ \cdot  JE }{ AK \cdot  KF}$$.

2010 Thailand Geometry Shortlist


Let $ABC$ be an isosceles triangle with $AB=AC$. Points $D$ and $E$ lie on sides $AB$ and $AC$ respectively, such that $\angle CDE=\angle ABC$. From point $D$, draw $DF\parallel BC$, that intersects $AC$ at point $F$. From point $E$, draw $EG\parallel CB$, that intersects $AB$ at point $G$. Prove that $DE^2= BG\cdot EF$

Let $\vartriangle ABC$ be an equilateral triangle, and let $M$ and $N$ be points on $AB$ and $AC$, respectively, so that $AN = BM$ and $3MB = AB$. Lines $CM$ and $BN$ intersect at $O$. Find $\angle AOB$.

Let $ABC$ be any triangle with point $D$ on side $BC$ such that $BD=\frac12 CD$. If the lengths of sides $BC$, $AC$ and $AB$ are equal to $a,b$ and $c$ respectively, then prove that$$|AD|^2 = \frac13 \left(2c^2+b^2-\frac23 a \right)$$

Let $\vartriangle ABC$ be an isosceles triangle with $AB = AC$. A circle passing through $B$ and $C$ intersects sides $AB$ and $AC$ at $D$ and $E$ respectively. A point $F$ on this circle is chosen so that $EF\perp  BC$. If $BC = x$, $CF = y$, and $BF = z$, find the length of $DF$ in terms of $x, y, z$.

Let $ABC$ be an acute triangle with altitude $CD$. Points $E$ and $F$ lie on line segments $AD$ and $BC$ respectively such that $\angle ECA=\angle BAF=15^o$. Let $AF$ intersect $CE$ and $CD$ at points $G$ and $H$ respectively. If triangles $AEH$ and $GFC$ are isosceles triangles with vertices at points $E$ and $G$ respectively, find the ratio between the area of triangle $EDH$ to the area of triangle $BCD$.

Let $ABCD$ be a quadrilateral with perpendicular diagonals and $\angle B=\angle D=90^o$. Let the circle of center $O$ be inscribed in the quadrilateral $ABCD$, with $M$ and $N$ the touchpoints with side $AB$ and $BC$ respectively. Let C be the center of a circle tangent to $AB$ and $AD$ at points $B$ and $D$ respectively. On extensions of $AB$ and $AD$ beyond $B$ and $D$, lie points $B'$ and $D'$ respectively, such that quadrilateral $AB'C'D'$ is similar to quadrilateral $ABCD$ and has incircle the circle of center $C$. In a quadrilateral $AB'C'D$',$N'$ is the touchpoints of it's incircle with sides $BC$. If $MN'$ is parallel to $AC'$, what is the ratio of $AB$ to $BC$, when $AB$ is longer than $BC$?

Let $\vartriangle ABC$ be a scalene triangle with $AB < BC < CA$. Let $D$ be the projection of $A$ onto the angle bisector of $\angle ABC$, and let $E$ be the projection of $A$ onto the angle bisector of $\angle ACB$. The line $DE$ cuts sides $AB$ and AC at $M$ and $N$, respectively. Prove that$$\frac{AB+AC}{BC} =\frac{DE}{MN} + 1$$

Let $ABC$ be an acute triangle with $AB>AC$. Point $D$ is lies on line segment $BC$, differs from $C$, such that $AC=AD$. Let $H$ be the orthocenter of triangle $ABC$. $A'$ and $B'$ are the feet of the altitudes drawn from points $A$ and $B$ respectively. Let the line $DH$ intersect $AC$ and $A'B'$ at points $E$ and $F$, respectively, and point $G$ is the intersection of lines $AF$ and $BH$. Prove that $CE=DT$ where $T$ is the intersection of $GE$ and $AD$.


2011 Thailand Geometry Shortlist

Let ABC be an isosceles triangle with $AB = AC$ and altitude $BD$. If $CD: AD = 1: 2$ , prove that $BC^2 = \frac23 AC^2$

Let $ABC$ be a triangle with points $D, E$, and $F$ on the line segments $AB, AC$, and $BC$ respectively such that $BFED$ is a paraellelogram. Points $H,G$ lie n the extenstions of DE ,DF beyond E, F respectively such that $HE: GE: FE = 1: 5: 8$, $HG = GF$ and $\angle BDF = \angle DHG$. Find the ratio $HE: HG$.

Let $ABCD$ be an inscribed quadrilateral , such that the diameter of the circumcircle $AC$ has length $10$ units. The diagonals $AC$ and $BD$ intersects at point $M$ and the length of $AM$ is $4$ units. Let the line $XY$ be the tangent line of the circle at point $A$. Extensions of sides $CD$ and $CB$ intersect the line $XY$ at the points $P$ and $Q$ respectively. Find the value of $AP \cdot  AQ$.

Given $\Delta ABC$ and its centroid $G$, If line $AC$ is tangent to $\odot (ABG)$. Prove that,\begin{align*} AB+BC \leq 2AC \end{align*}

Given triangle $ABC$ , points $D$ and $F$ lie on sides $BC$ and $AB$ respectively, such that $BD = 7$, $DC = 2$, $BF = 5$, $FA = 2$ and $AD$ intersects $FC$ at point $P$. If $PC = 3$, then find the lengths of $AP$ and $PD$.

A quadrilateral $ABCD$ inscribed in a circle. Let $M$ and $N$ be the midpoints of sides $AB$ and $CD$ . If the diagonal $AC$, the diagonal $BD$ and the line $MN$ intersect at one point, then angle $BAD$ is equal to angle $ABC$.

Gives a triangle $ABC$ inscribed in a circle of radius $5$ units. Let O be it's orthocenter. If the side $BC$ is $8$ units, then find the length of $AO$.

Given a $\Delta ABC$ where $\angle C = 90^{\circ}$, $D$ is a point in the interior of $\Delta ABC$ and lines $AD$ $,$ $BD$ and $CD$ intersect $BC$, $CA$ and $AB$ at points $P$ ,$Q$ and $R$ ,respectively. Let $M$ be the midpoint of $\overline{PQ}$. Prove that, if $\angle BRP$ $ =$ $ \angle PRC$ then $MR=MC$.

Let $I$ be the center of the inscribed circle of triangle $ABC$ and $AI, BI, CI$ intersect the sides $BC, CA, AB$ at points $A_1, B_1, C_1$, respectively. Prove that$$\frac{AI}{A_1I} \frac{BI}{B_1I} \frac{CI}{C_1I}\ge 8.$$

Let $ABC$ be a triangle where angles $ABC$ and $BAC$ are acute. The bisector of internal and external angles of angle $BAC$ intersect the line $BC$ at points $D$ and $E$, respectively. Let $O$ be the center of the circumcscribed circle of the triangle $ADE$. If point $P$ lie on this circle of center $O$, prove that$$\frac{BP}{PC}=\frac{OB}{OA}$$

Circles of radii $r_1, r_2$ and $r_3$ are externally touching each other at points $A, B$, and $C$. If the triangle $ABC$ has perimeter equal to $p$, prove that$$\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}\ge \frac{9}{p}.$$

In $\Delta ABC$, Let the Incircle touch $\overline{BC}, \overline{CA}, \overline{AB}$ at $X,Y,Z$. Let $I_A,I_B,I_C$ be $A$,$B$,$C-$excenters, respectively. Prove that Incenter of $\Delta ABC$, orthocenter of $\Delta XYZ$ and centroid of $\Delta I_AI_BI_C$ are collinear.

2012 Thailand Geometry Shortlist

1Let $ABC$ be a right triangle with $\angle BAC$ right angle and $AB = \frac12 BC$. Let $D$ be the midpoint of $BC$ and $E$ be the point on the same semiplane with $A$ wrt the line $BC$ such that $DE = AB$. From point $E$ the perpendicular on $AC$ cuts it at point $F$. Line $DF$ intersects $AE$ at point $G$. Prove that $GD$ is perpendicular to $AE$

Let $\vartriangle ABC$ be a right triangle with $\angle B = 90^o$. Let $P$ be a point on side $BC$, and let $\omega$ be the circle with diameter $CP$. Suppose that $\omega$ intersects $AC $and $AP$ again at $Q$ and $R$, respectively. Show that $CP^2 = AC \cdot CQ - AP \cdot P R$.

Let $ABC$ be a triangle with $\angle ABC = 45^o$. Point $P$ lies on the side BC that $PC=2$ units and $\angle BAP = 15^o$. If the line segment of the tangent drawn from point $B$ to the circumcircle of the triangle $APC$ is $\sqrt3$ units, find the measure of the $\angle ACP$.

Let $ABCD$ be an cyclic quadrilateral. Let the diagonal $AC$ and $BD$ meet at point $X$. Let $Z$ be point on $AD$ such that $AZ = ZX$. The line $ZX$ intersects the side $BC$ at the point $Y$. If $AD^2 = 2DX^2$ , prove that $BY = YC$.

Let $ABCD$ be a unit square. Points $E, F, G, H$ are chosen outside $ABCD$ so that $\angle AEB =\angle BF C = \angle CGD = \angle DHA = 90^o$ . Let $O_1, O_2, O_3, O_4$, respectively, be the incenters of $\vartriangle ABE, \vartriangle BCF, \vartriangle CDG, \vartriangle DAH$. Show that the area of $O_1O_2O_3O_4$ is at most $1$.

Let $A_1A_2A_3$ be a triangle with incenter $I$. Let the inner angle bisectors of $ \angle A_1, \angle A_2, \angle A_3$ intersect the circumcircle again at $B_1, B_2, B_3$ respectively. Prove that$$\frac{IA_i^2}{IB_i}\ge 2r$$for any $i\in \{1,2,3\}$ , where $r$ is the radius of the inscribed circle of the triangle $A_1A_2A_3$ .

Let $ABC$ be a triangle, with $I$ the center of the inscribed circle, with touches sides $AB$ and $BC$ at points $D$ and $E$ respectively. Let $K$ and $L$ be points on the incircle, such that $DK$ and $EL$ are diameters of the incircle. If $AB + BC = 3AC$, prove that $A,C,I,K,L$ lie on the same circle.
Let $ABC$ be a triangle with $AC> BC$. A circle that passes through point $A$ and touches the side $BC$ at point $B$, intersects line $AC$ at $D\ne A$. Line $BD$ intersects the circumcircle of triangle $ABC$ again at point $E$. Let $F$ be the point on the circumscribed circle of triangle $CDE$ such that $DF = DB$. Prove that $F$ lies on the line $AE$ or $BC$.

Let $\ell$ be the common tangent of $\omega_1$ and $\omega_2$ which is tangent at $\omega_1$ and $\omega_2$ at points $A$ and $B$ respectively where the circles $\omega_1$ and $\omega_2$ lie on the same side wrt $\ell$ . Let $M$ be the midpoint of $AB$. From point $M$ draw a tangent to $\omega_1$ that intersects it at point $C\ne A$. From point $M$ draw a tangent to $\omega_2$ that intersects it at point $D\ne B$. Let $P$ be a point on $O_1O_2$ such that $MP \perp O_1O_2 $. Show that the circumscribed circle of the triangle $CPD$ is tangent to the circles $\omega_1$ and $\omega_2$.

Let $\vartriangle ABC$ be an acute triangle, and let $P$ be the foot of altitude from $C$ to $AB$. Let $\omega$ be the circle with diameter $BC$. The tangents from $A$ to $\omega$ are drawn touching $\omega$ at $D$ and $E$. Lines $AD$ and $AE$ intersect line $BC$ at $M$ and $N$ respectively, so that $B$ lies between $M$ and $C$. Let $CP$ intersect $DE$ at $Q, ME$ intersect $ND$ at $R$, and let $QR$ intersect $BC$ at $S$. Show that $QS$ bisects $\angle DSE$

Let the line segment $AB$ be a common chord of two circles with center $O_1$ and $O_2$ ($O_1 \ne O_2$). Let $k_1$ and $k_2$ represent the arc $AB$ on the same side wrt line $AB$ of the circle $O_1$ and $O_2$ respectively, with $k_1$ between $k_2$ and $AB$. Let $X$ be the point on $k_2$ that does not lie on the perpendicular bisector of the segment $AB$. Tangent at $X$ of $k_2$ intersects $AB$ at point $C$. Let $Y$ be a point on $k_1$ such that $CX = CY$. Show that the line $XY$ passes through a fixed point independent of the position of $X$ on $k_2$.


2014 Thailand Geometry Shortlist

Let $ABCD$ be a square with side $1$, with $P$ and $Q$ being points on the sides $AB$ and $BC$, respectively, such that $PB + BQ = 1$. If $PC$ intersects $AQ$ at $E$, prove that the line $DE$ is perpendicular to the line $PQ$.

$ABC$ is an acute triangle with $D, E$ and $F$ being the feet of the altitudes of the triangle $ABC$ on sides $BC, AC$ and $AB$ respectively. Let $P, Q$ and $R$ be the midpoints of $DE, EF$ and $FD$ respectively. Then show that the lines passing through $P, Q$, and $R$ perpendicular on sides $AB, BC$, and $CA$, respectively, intersect at a single point.

Let $\vartriangle ABC$ be an isosceles triangle with $\angle BAC = 100^o$. Let $D, E$ be points on ray $\overrightarrow{AB}$ so that $BC = AD = BE$. Show that $BC \cdot  DE = BD \cdot CE$

Let $ABC$ be an isosceles triangle with $A$ being the apex, less than $60^o$ with $D$ the point on the side $AC$ , such that $\angle DBC =  \angle BAC$. Let $L_1$ be a line passing through point $A$ and parallel to side $BC$. Let $L_2$ be the perpendicular bisector of side $BD$. $L_1$ and $L_2$ intersect at point $E$. show that the $EC$ is bisected by $AB$.

Let $ABC$ be a triangle with circumcircle $\Gamma$. Let the tangents of circle $\Gamma$ at points $B$ and $C$ intersect at point $D$ . Let $M$ be the point on the side $BC$ such that $\angle BAM = \angle CAD$. Prove that the center of circle $\Gamma$ lies on the line $MD$.

Let A$BC$ be a triangle with $A$ right angle and $D$ is a point on the side $BC$ such that $AD$ is perpendicular to the side $BC$. Let $W_1$ and $W_2$ are the centers of the incircles of the triangles $ABD$ and $ADC$ respectively. Line $W_1W_2$ intersects $AB$ at $X$ and $AC$ at $Y$. Prove that $AX = AD = AY$.

Let $ABC$ be an acute triangle with $E$ and $F$ on sides $AB$ and $AC$ respectively, and $O$ be it's circumcenter. Let $AO$ intersect $BC$ at point $D$. Let the perpendicular from point $D$ on sides $AB$ and $AC$ intersect them at points $M,N$ respectively. Let the perpendicular on the side $BC$ from points $E,F,M,N$ intersect it at points $E',F',M',N'$ respectively. Prove that $A, D, E, F$ lie on the same circle if and only if $E'F '= M'N'$.

Let $ABCD$ be a convex quadrilateral with shortest side $AB$ and longest side $CD$, and suppose that $AB < CD$. Show that there is a point $E \ne C, D$ on segment $CD$ with the following property:
For all points $P \ne E$ on side $CD$, if we define $O_1$ and $O_2$ to be the circumcenters of $\vartriangle APD$ and $\vartriangle  BPE$ respectively, then the length of $O_1O_2$ does not depend on $P$.

2014 Indonesia INAMO Geometry Shortlist

The inscribed circle of the $ABC$ triangle has center  $I$ and touches to $BC$ at $X$. Suppose the $AI$ and $BC$ lines intersect at $L$, and $D$ is the reflection of $L$ wrt $X$. Points $E$ and $F$ respectively are the result of a reflection of $D$ wrt to lines $CI$ and $BI$ respectively. Show that quadrilateral $BCEF$ is cyclic .

2014 INAMO Shortlist G2 (problem 6)
Let $ABC$ be a triangle. Suppose $D$ is on $BC$ such that $AD$ bisects $\angle BAC$. Suppose $M$ is on $AB$ such that $\angle MDA = \angle ABC$, and $N$ is on $AC$ such that $\angle NDA = \angle ACB$. If $AD$ and $MN$ intersect on $P$, prove that $AD^3 = AB \cdot AC \cdot AP$.

2014 INAMO Shortlist G3 (problem 3)
Let $ABCD$ be a trapezoid (quadrilateral with one pair of parallel sides) such that $AB < CD$. Suppose that $AC$ and $BD$ meet at $E$ and $AD$ and $BC$ meet at $F$. Construct the parallelograms $AEDK$ and $BECL$. Prove that $EF$ passes through the midpoint of the segment $KL$.

2014 INAMO Shortlist G4
Given an acute triangle $ABC$ with $AB <AC$. Points $P$ and $Q$ lie on the angle bisector of $\angle BAC$ so that $BP$ and $CQ$ are perpendicular on that angle bisector. Suppose that point $E, F$ lie respectively at sides $AB$ and $AC$ respectively, in such a way that $AEPF$ is a kite. Prove that the lines $BC, PF$, and $QE$ intersect at one point.

2014 INAMO Shortlist G5
Given a cyclic quadrilateral $ABCD$. Suppose $E, F, G, H$ are respectively the midpoint of the sides $AB, BC, CD, DA$. The line passing through $G$ and perpendicular on $AB$ intersects the line passing through $H$ and perpendicular on $BC$ at point $K$. Prove that $\angle EKF = \angle  ABC$.

2014 INAMO Shortlist G6
Given an $ABC$ acute triangle with $O$ the center of the circumscribed circle. Suppose that $\omega$ is a circle that is tangent to the line $AO$ at point $A$ and also tangent to the line $BC$. Prove that  $\omega$ is also tangent to the circumcircle of the triangle $BOC$.

2015 Indonesia INAMO Geometry Shortlist

2015 INAMO Shortlist G1
Given a cyclic quadrilateral $ABCD$ so that $AB = AD$ and $AB + BC <CD$. Prove that the angle $ABC$ is more than $120$ degrees.

2015 INAMO Shortlist G2
Two circles that are not equal are tangent externally at point $R$. Suppose point $P$ is the intersection of the external common tangents of the two circles. Let $A$ and $B$ are two points on different circles so that $RA$ is perpendicular to $RB$. Show that the line $AB$ passes through $P$.
2015 INAMO Shortlist G3
Given $ABC$ triangle with incircle $L_1$ and circumcircle $L_2$. If points $X, Y, Z$ lie on $L_2$, such that $XY, XZ$ are tangent to $L_1$, then prove that $YZ$ is also tangent to $L_1$.

2015 INAMO Shortlist G4
Given an isosceles triangle $ABC$ with $AB = AC$, suppose $D$ is the midpoint of the $AC$. The circumcircle of the $DBC$ triangle intersects the altitude from $A$ at point $E$ inside the triangle  $ABC$, and the circumcircle of the triangle $AEB$ cuts the side  $BD$ at point $F$. If $CF$ cuts $AE$ at point $G$, prove that $AE = EG$.

2015 INAMO Shortlist G5 
Let $ABC$ be an acute triangle. Suppose that circle $\Gamma_1$ has it's center on the side $AC$ and is tangent to the sides $AB$ and $BC$, and circle $\Gamma_2$ has it's center on the side  $AB$ and is tangent to the sides $AC$ and $BC$. The circles  $\Gamma_1$ and $ \Gamma_2$ intersect at two points $P$ and $Q$. Show that if $A, P, Q$ are collinear, then $AB = AC$.

2015 INAMO Shortlist G6 (problem 6)
Let $ABC$ be an acute angled triangle with circumcircle $O$. Line $AO$ intersects the circumcircle of triangle  $ABC$ again at point $D$. Let $P$ be a point on the side  $BC$. Line passing through $P$ perpendicular to $AP$ intersects lines  $DB$ and $DC$ at $E$ and $F$ respectively . Line passing through $D$ perpendicular to $BC$ intersects $EF$ at point $Q$. Prove that $EQ = FQ$ if and only if $BP = CP$.

2015 INAMO Shortlist G7 (problem 3)
Given an acute triangle $ABC$. $\Gamma _{B}$ is a circle that passes through $AB$, tangent to $AC$ at $A$ and centered at $O_{B}$. Define $\Gamma_C$ and $O_C$ the same way. Let the altitudes of $\triangle ABC$ from $B$ and $C$ meets the circumcircle of $\triangle ABC$ at $X$ and $Y$, respectively. Prove that $A$, the midpoint of $XY$ and the midpoint of $O_{B}O_{C}$ is collinear.

$ABC$ is an acute triangle with $AB> AC$.  $\Gamma_B$ is a circle that passes through $A,B$ and is tangent to $AC$ on $A$. Define similar for  $ \Gamma_C$. Let $D$ be the intersection  $\Gamma_B$ and  $\Gamma_C$ and $M$ be the midpoint of $BC$. $AM$ cuts  $\Gamma_C$ at $E$. Let $O$ be the center of the circumscibed circle of the triangle ABC. Prove that the circumscibed circle of the triangle $ODE$ is tangent to $\Gamma_B$.

                       2015 Junior Olympiad of Malaysia
                                    Geometry Shortlist 

Given a triangle $ABC$, and let $ E $ and $ F $ be the feet of altitudes from vertices $ B $ and $ C $ to the opposite sides. Denote $ O $ and $ H $ be the circumcenter and orthocenter of triangle $ ABC $. Given that $ FA=FC $, prove that $ OEHF $ is a parallelogram.

Let $ ABC $ be a triangle, and let $M$ be midpoint of $BC$. Let $ I_b $ and $ I_c $ be incenters of $ AMB $ and $ AMC $. Prove that the second intersection of circumcircles of $ ABI_b $ and $ ACI_c $ distinct from $A$ lies on line $AM$.

Let $ ABC$ a triangle. Let $D$ on $AB$ and $E$ on $AC$ such that $DE||BC$. Let line $DE$ intersect circumcircle of $ABC$ at two distinct points $F$ and $G$ so that line segments $BF$ and $CG$ intersect at P. Let circumcircle of $GDP$ and $FEP$ intersect again at $Q$. Prove that $A, P, Q$ are collinear.

Let $ ABC $ be a triangle and let $ AD, BE, CF $ be cevians of the triangle which are concurrent at $ G $. Prove that if $ CF \cdot BE \ge AF \cdot EC + AE \cdot BF + BC \cdot FE $ then $ AG \le GD $.

Let $ ABCD $ be a convex quadrilateral. Let angle bisectors of $ \angle B $ and $ \angle C $ intersect at $ E $. Let $ AB $ intersect $ CD $ at $ F $. Prove that if $ AB+CD=BC $, then $A,D,E,F$ are concyclic.

Let $ABC$ be a triangle. Let $\omega_1$ be circle tangent to $BC$ at $B$ and passes through $A$. Let $\omega_2$ be circle tangent to $BC$ at $C$ and passes through $A$. Let $\omega_1$ and $\omega_2$ intersect again at $P \neq A$. Let $\omega_1$ intersect $AC$ again at $E\neq A$, and let $\omega_2$ intersect $AB$ again at $F\neq A$. Let $R$ be the reflection of $A$ about $BC$, Prove that lines $BE, CF, PR$ are concurrent.

Let $ABC$ be an acute triangle. Let $H_A,H_B,H_C$ be points on $BC,AC,AB$ respectively such that $AH_A\perp BC, BH_B\perp AC, CH_C\perp AB$. Let the circumcircles $AH_BH_C,BH_AH_C,CH_AH_B$ be $\omega_A,\omega_B,\omega_C$ with circumcenters $O_A,O_B,O_C$ respectively and define $O_AB\cap \omega_B=P_{AB}\neq B$. Define $P_{AC},P_{BA},P_{BC},P_{CA},P_{CB}$ similarly. Define circles $\omega_{AB},\omega_{AC}$ to be $O_AP_{AB}H_C,O_AP_{AC}H_B$ respectively. Define circles $\omega_{BA},\omega_{BC},\omega_{CA},\omega_{CB}$ similarly.
Prove that there are $6$ pairs of tangent circles in the $6$ circles of the form $\omega_{xy}$.

Let $ ABCDE $ be a convex pentagon such that $ BC $ and $ DE $ are tangent to the circumcircle of $ ACD $. Prove that if the circumcircles of $ ABC $ and $ ADE $ intersect at the midpoint of $ CD $, then the circumcircles $ ABE $ and $ ACD $ are tangent to each other.

2016 Romanian Master Of Mathematics
Geometry Shortlist 

RMM  2016 SHL G1 (also Romania TST1 p1 2016)
Two circles,  $\omega_1$ and $\omega_2$, centred at $O1$ and $O2$, respectively, meet at points $A$ and $B$. A line through $B$ meets $\omega_1$ again at $C$, and $\omega_2$ again at $D$. The tangents to  $\omega_1$ and $\omega_2$ at $C$ and $D$, respectively, meet at $E$, and the line $AE$ meets the circle $\omega$ through $A, O_1, O_2$ again at $F$. Prove that the length of the segment $EF$ is equal to the diameter of $\omega$.


2017 Indonesia INAMO Geometry Shortlist

Let $ABCD$ be a square with side $1$, with $P$ and $Q$ being points on the sides $AB$ and $BC$, respectively, such that $PB + BQ = 1$. If $PC$ intersects $AQ$ at $E$, prove that the line $DE$ is perpendicular to the line $PQ$.

$ABC$ is an acute triangle with $D, E$ and $F$ being the feet of the altitudes of the triangle $ABC$ on sides $BC, AC$ and $AB$ respectively. Let $P, Q$ and $R$ be the midpoints of $DE, EF$ and $FD$ respectively. Then show that the lines passing through $P, Q$, and $R$ perpendicular on sides $AB, BC$, and $CA$, respectively, intersect at a single point.

Let $\vartriangle ABC$ be an isosceles triangle with $\angle BAC = 100^o$. Let $D, E$ be points on ray $\overrightarrow{AB}$ so that $BC = AD = BE$. Show that $BC \cdot  DE = BD \cdot CE$

Let $ABC$ be an isosceles triangle with $A$ being the apex, less than $60^o$ with $D$ the point on the side $AC$ , such that $\angle DBC =  \angle BAC$. Let $L_1$ be a line passing through point $A$ and parallel to side $BC$. Let $L_2$ be the perpendicular bisector of side $BD$. $L_1$ and $L_2$ intersect at point $E$. show that the $EC$ is bisected by $AB$.

Let $ABC$ be a triangle with circumcircle $\Gamma$. Let the tangents of circle $\Gamma$ at points $B$ and $C$ intersect at point $D$ . Let $M$ be the point on the side $BC$ such that $\angle BAM = \angle CAD$. Prove that the center of circle $\Gamma$ lies on the line $MD$.

Let A$BC$ be a triangle with $A$ right angle and $D$ is a point on the side $BC$ such that $AD$ is perpendicular to the side $BC$. Let $W_1$ and $W_2$ are the centers of the incircles of the triangles $ABD$ and $ADC$ respectively. Line $W_1W_2$ intersects $AB$ at $X$ and $AC$ at $Y$. Prove that $AX = AD = AY$.

Let $ABC$ be an acute triangle with $E$ and $F$ on sides $AB$ and $AC$ respectively, and $O$ be it's circumcenter. Let $AO$ intersect $BC$ at point $D$. Let the perpendicular from point $D$ on sides $AB$ and $AC$ intersect them at points $M,N$ respectively. Let the perpendicular on the side $BC$ from points $E,F,M,N$ intersect it at points $E',F',M',N'$ respectively. Prove that $A, D, E, F$ lie on the same circle if and only if $E'F '= M'N'$.

Let $ABCD$ be a convex quadrilateral with shortest side $AB$ and longest side $CD$, and suppose that $AB < CD$. Show that there is a point $E \ne C, D$ on segment $CD$ with the following property:
For all points $P \ne E$ on side $CD$, if we define $O_1$ and $O_2$ to be the circumcenters of $\vartriangle APD$ and $\vartriangle  BPE$ respectively, then the length of $O_1O_2$ does not depend on $P$.

2017 Romanian Master Of Mathematics
Geometry Shortlist

RMM  2017 SHL G1 (also Romania TST1 p1 2017)
Let $ABCD$ be a trapezium, $AD\parallel BC$, and let $E,F$ be points on the sides$AB$ and $CD$, respectively. The circumcircle of $AEF$ meets $AD$ again at $A_1$, and the circumcircle of $CEF$ meets $BC$ again at $C_1$. Prove that $A_1C_1,BD,EF$ are concurrent.

by Alexander Kuznetsov, Russia

Let $ABC$ be a triangle. Consider the circle $\omega_B$ internally tangent to the sides $BC$ and $BA$, and to the circumcircle of the triangle $ABC$, let $P$ be the point of contact of the two circles. Similarly, consider the circle $\omega_C$ internally tangent to the sides $CB$ and $CA$, and to the circumcircle of the triangle $ABC$, let $Q$ be the point of contact of the two circles. Show that the incentre of the triangle $ABC$ lies on the segment $PQ$ if and only if $AB + AC = 3BC$.
by Luis Eduardo Garcia Hernandez, Mexico

RMM  2017 SHL G3 (also Romania TST3 p4 2017)
Let $ABCD$ be a convex quadrilateral and let $P$ and $Q$ be variable points inside this quadrilateral so that $\angle APB=\angle CPD=\angle AQB=\angle CQD$. Prove that the lines $PQ$ obtained in this way all pass through a fixed point , or they are all parallel.
by Sergey Berlov, Russia

2018 Romanian Master Of Mathematics
Geometry Shortlist
Let $ABC$ be a triangle and let $H$ be the orthogonal projection of $A$ on the line $BC$. Let $K$ be a point on the segment $AH$ such that $AH = 3 KH$. Let $O$ be the circumcenter of triangle $ABC$ and let $M$ and $N$ be the midpoints of sides $AC$ and $AB$ respectively. The lines $KO$ and $MN$ meet at a point $Z$ and the perpendicular at $Z$ to $OK$ meets lines $AB, AC$ at $X$ and $Y$ respectively. Show that $\angle XKY = \angle CKB$.

by Italy
Let $\triangle ABC$ be a triangle, and let $S$ and $T$ be the midpoints of the sides $BC$ and $CA$, respectively. Suppose $M$ is the midpoint of the segment $ST$ and the circle $\omega$ through $A, M$ and $T$ meets the line $AB$ again at $N$. The tangents of $\omega$ at $M$ and $N$ meet at $P$. Prove that $P$ lies on $BC$ if and only if the triangle $ABC$ is isosceles with apex at $A$.

by Reza Kumara, Indonesia


2018 Pan African MO Geometry Shortlist

PAMO Shortlist 2018  G1
In a triangle $ABC$, let $D$ and $E$ be the midpoints of $AB$ and $AC$, respectively, and let $F$ be the foot of the altitude through $A$. Show that the line $DE$, the angle bisector of $\angle ACB$ and the circumcircle of $ACF$ pass through a common point.

 Alternate version: In a triangle $ABC$, let $D$ and $E$ be the midpoints of $AB$ and $AC$, respectively. The line $DE$ and the angle bisector of $\angle ACB$ meet at $G$. Show that $\angle AGC$ is a right angle.

PAMO Shortlist 2018  G2
Let $P$ be a point on the median $AM$ of a triangle $ABC$. Suppose that the tangents to the circumcircles of $ABP$ and $ACP$ at $B$ and $C$ respectively meet at $Q$. Show that $\angle PAB = \angle CAQ$.

PAMO Shortlist 2018  G3 (problem 4)
Given a triangle $ABC$, let $D$ be the intersection of the line through $A$ perpendicular to $AB$, and the line through $B$ perpendicular to $BC$. Let $P$ be a point inside the triangle. Show that $DAPB$ is cyclic if and only if $\angle BAP = \angle CBP$.

PAMO Shortlist 2018  G4
Let $ABC$ be a triangle and $\Gamma$ be the circle with diameter $[AB]$. The bisectors of $\angle BAC$ and $\angle ABC$ cut the circle $\Gamma$ again at $D$ and $E$, respectively. The incicrcle of the triangle $ABC$ cuts the lines $BC$ and $AC$ in $F$ and $G$, respectively. Show that the points $D, E, F$ and $G$ lie on the same line.

PAMO Shortlist 2018  G5
Let $ABC$ be a triangle with $AB \neq AC$. The incircle of $ABC$ touches the sides $BC$, $CA$, $AB$ at $X$, $Y$, $Z$ respectively. The line through $Z$ and $Y$ intersects $BC$ extended in $X^\prime$. The lines through $B$ that are parallel to $AX$ and $AC$ intersect $AX^\prime$ in $K$ and $L$ respectively. Prove that $AK = KL$.

PAMO Shortlist 2018  G6
Let $\Gamma$ be the circumcircle of an acute triangle $ABC$. The perpendicular line to $AB$ passing through $C$ cuts $AB$ in $D$ and $\Gamma$ again in $E$. The bisector of the angle $C$ cuts $AB$ in $F$ and $\Gamma$ again in $G$. The line $GD$ meets $\Gamma$ again at $H$ and the line $HF$ meets $\Gamma$ again at $I$. Prove that $AI = EB$.
2019 Romanian Master Of Mathematics
                                          Geometry Shortlist
 
Let $BM$ be a median in an acute-angled triangle $ABC$. A point $K$ is chosen on the line through $C$ tangent to the circumcircle of $\vartriangle BMC$ so that $\angle KBC = 90^o$. The segments $AK$ and $BM$ meet at $J$. Prove that the circumcenter of $\vartriangle BJK$ lies on the line $AC$.

Aleksandr Kuznetsov, Russia
Let $ABC$ be an acute-angled triangle. The line through $C$ perpendicular to $AC$ meets the external angle bisector of $\angle ABC$ at $D$. Let $H$ be the foot of the perpendicular from $D$ onto $BC$. The point $K$ is chosen on $AB$ so that $KH \parallel AC$. Let $M$ be the midpoint of $AK$. Prove that $MC = MB + BH$.

Giorgi Arabidze, Georgia
RMM  2019 SHL G3 (EMC 2019 P3)
Let $ABC$ be an acute-angled triangle with $AB \ne AC$, and let $I$ and $O$ be its incenter and circumcenter, respectively. Let the incircle touch $BC, CA$ and $AB$ at $D, E$ and $F$, respectively. Assume that the line through $I$ parallel to $EF$, the line through $D$ parallel to$ AO$, and the altitude from $A$ are concurrent. Prove that the concurrency point is the orthocenter of the triangle $ABC$.

Petar Nizic-Nikolac, Croatia
Let $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. Let $D$ be the midpoint of the minor arc $AB$ of $\Omega$. A circle $\omega$ centered at $D$ is tangent to $AB$ at $E$. The tangents to $\omega$ through $C$ meet the segment $AB$ at $K$ and $L$, where $K$ lies on the segment $AL$. A circle $\Omega_1$ is tangent to the segments $AL, CL$, and also to $ \Omega$ at point $M$. Similarly, a circle $\Omega_2$ is tangent to the segments $BK, CK$, and also to $\Omega$ at point $N$. The lines $LM$ and $KN$ meet at $P$. Prove that $\angle KCE = \angle LCP$.

Poland
Let $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. A point $D$ is chosen on the internal bisector of $\angle ACB$ so that the points $D$ and $C$ are separated by $AB$. A circle $\omega$ centered at $D$ is tangent to the segment $AB$ at $E$. The tangents to $\omega$ through $C$ meet the segment $AB$ at $K$ and $L$, where $K$ lies on the segment $AL$. A circle $\Omega_1$ is tangent to the segments $AL, CL$, and also to $\Omega$ at point $M$. Similarly, a circle $\Omega_2$ is tangent to the segments $BK, CK$, and also to $\Omega$ at point $N$. The lines $LM$ and $KN$ meet at $P$. Prove that $\angle KCE = \angle LCP$.

Poland
A quadrilateral $ABCD$ is circumscribed about a circle with center $I$. A point $P \ne I$ is chosen inside $ABCD$ so that the triangles $PAB, PBC, PCD,$ and $PDA$ have equal perimeters. A circle $\Gamma$ centered at $P$ meets the rays $PA, PB, PC$, and $PD$ at $A_1, B_1, C_1$, and $D_1$, respectively. Prove that the lines $PI, A_1C_1$, and $B_1D_1$ are concurrent.


Ankan Bhattacharya, US
RMM 2019 Original P4 (removed due to leak) 
Let there be an equilateral triangle $ABC$ and a point $P$ in its plane such that $AP<BP<CP.$ Suppose that the lengths of segments $AP,BP$ and $CP$ uniquely determine the side of $ABC$. Prove that $P$ lies on the circumcircle of triangle $ABC.$


1 comment:

  1. Rum TanApril 29, 2024 at 9:07 PM

    Geometry in mathematics deals with the study of shapes, sizes, properties, and dimensions of objects in space. It encompasses concepts such as points, lines, angles, surfaces, and solids, analyzing their relationships and characteristics. Through geometry, we explore spatial arrangements, measure distances, calculate areas, volumes, and angles, and solve problems involving transformations like rotations, reflections, and translations. It serves as a foundation for understanding the physical world, engineering, architecture, and various scientific disciplines, providing tools to describe and analyze the structure and arrangement of objects.

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