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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 $ABC.$


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