IGO 2022

Find the angles of the pentagon $ABCDE$ in the figure below.

Attachments:

An isosceles trapezoid $ABCD$ $(AB \parallel CD)$ is given. Points $E$ and $F$ lie on the sides $BC$ and $AD$ , and the points $M$ and $N$ lie on the segment $EF$ such that $DF = BE$ and $FM = NE$ . Let $K$ and $L$ be the foot of perpendicular lines from $M$ and $N$ to $AB$ and $CD$ , respectively. Prove that $EKFL$ is a parallelogram.

Proposed by Mahdi Etesamifard

Let $ABCDE$ be a convex pentagon such that $AB = BC = CD$ and $\angle BDE = \angle EAC = 30 ^{\circ}$ . Find the possible values of $\angle BEC$ .

Proposed by Josef Tkadlec (Czech Republic)

Let $AD$ be the internal angle bisector of triangle $ABC$ . The incircles of triangles $ABC$ and $ACD$ touch each other externally. Prove that $\angle ABC > 120^{\circ}$ . (Recall that the incircle of a triangle is a circle inside the triangle that is tangent to its three sides.)

Proposed by Volodymyr Brayman (Ukraine)

a) Do there exist four equilateral triangles in the plane such that each two have exactly one vertex in common, and every point in the plane lies on the boundary of at most two of them? b) Do there exist four squares in the plane such that each two have exactly one vertex in common, and every point in the plane lies on the boundary of at most two of them? (Note that in both parts, there is no assumption on the intersection of interior of polygons.)

Proposed by Hesam Rajabzadeh

Given is a circle $\omega$ and a line $\ell$ tangent to $\omega$ at $Y$ . Point $X$ lies on $\ell$ to the left of $Y$ . The tangent to $\omega$ , perpendicular to $\ell$ meets $\ell$ at $A$ and touches $\omega$ at $D$ . Let $B$ a point on $\ell$ , to the right of $Y$ , such that $AX=BY$ . The tangent from $B$ to $\omega$ touches the circle at $C$ . Prove that $\angle XDA= \angle YDC$ .

Note: This is not the official wording (it was just a diagram without any description).

Two circles $\omega_1$ and $\omega_2$ with equal radius intersect at two points $E$ and $X$ . Arbitrary points $C, D$ lie on $\omega_1, \omega_2$ . Parallel lines to $XC, XD$ from $E$ intersect $\omega_2, \omega_1$ at $A, B$ , respectively. Suppose that $CD$ intersect $\omega_1, \omega_2$ again at $P, Q$ , respectively. Prove that $ABPQ$ is cyclic.

Proposed by Ali Zamani

Let $O$ be the circumcenter of triangle $ABC$ . Arbitrary points $M$ and $N$ lie on the sides $AC$ and $BC$ , respectively. Points $P$ and $Q$ lie in the same half-plane as point $C$ with respect to the line $MN$ , and satisfy $\triangle CMN \sim \triangle PAN \sim \triangle QMB$ (in this exact order). Prove that $OP=OQ$ .

Proposed by Medeubek Kungozhin, Kazakhstan

We call two simple polygons $P, Q$ $\textit{compatible}$ if there exists a positive integer $k$ such that each of $P, Q$ can be partitioned into $k$ congruent polygons similar to the other one. Prove that for every two even integers $m, n \geq 4$ , there are two compatible polygons with $m$ and $n$ sides. (A simple polygon is a polygon that does not intersect itself.)

Proposed by Hesam Rajabzadeh

Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$ with center $O$ . Let $P$ be the intersection of two diagonals $AC$ and $BD$ . Let $Q$ be a point lying on the segment $OP$ . Let $E$ and $F$ be the orthogonal projections of $Q$ on the lines $AD$ and $BC$ , respectively. The points $M$ and $N$ lie on the circumcircle of triangle $QEF$ such that $QM \parallel AC$ and $QN \parallel BD$ . Prove that the two lines $ME$ and $NF$ meet on the perpendicular bisector of segment $CD$ .

Proposed by Tran Quang Hung, Vietnam

Four points $A$ , $B$ , $C$ and $D$ lie on a circle $\omega$ such that $AB=BC=CD$ . The tangent line to $\omega$ at point $C$ intersects the tangent line to $\omega$ at $A$ and the line $AD$ at $K$ and $L$ . The circle $\omega$ and the circumcircle of triangle $KLA$ intersect again at $M$ . Prove that $MA=ML$ .

Proposed by Mahdi Etesamifard

We are given an acute triangle $ABC$ with $AB\neq AC$ . Let $D$ be a point of $BC$ such that $DA$ is tangent to the circumcircle of $ABC$ . Let $E$ and $F$ be the circumcenters of triangles $ABD$ and $ACD$ , respectively, and let $M$ be the midpoints $EF$ . Prove that the line tangent to the circumcircle of $AMD$ through $D$ is also tangent to the circumcircle of $ABC$ .

Proposed by Patrik Bak, Slovakia

In triangle $ABC$ $(\angle A\neq 90^\circ)$ , let $O$ , $H$ be the circumcenter and the foot of the altitude from $A$ respectively. Suppose $M$ , $N$ are the midpoints of $BC$ , $AH$ respectively. Let $D$ be the intersection of $AO$ and $BC$ and let $H'$ be the reflection of $H$ about $M$ . Suppose that the circumcircle of $OH'D$ intersects the circumcircle of $BOC$ at $E$ . Prove that $NO$ and $AE$ are concurrent on the circumcircle of $BOC$ .

Proposed by Mehran Talaei

Let $ABCD$ be a trapezoid with $AB\parallel CD$ . Its diagonals intersect at a point $P$ . The line passing through $P$ parallel to $AB$ intersects $AD$ and $BC$ at $Q$ and $R$ , respectively. Exterior angle bisectors of angles $DBA$ , $DCA$ intersect at $X$ . Let $S$ be the foot of $X$ onto $BC$ . Prove that if quadrilaterals $ABPQ$ , $CDQP$ are circumcribed, then $PR=PS$ .

Proposed by Dominik Burek, Poland

Let $ABC$ be an acute triangle inscribed in a circle $\omega$ with center $O$ . Points $E$ , $F$ lie on its side $AC$ , $AB$ , respectively, such that $O$ lies on $EF$ and $BCEF$ is cyclic. Let $R$ , $S$ be the intersections of $EF$ with the shorter arcs $AB$ , $AC$ of $\omega$ , respectively. Suppose $K$ , $L$ are the reflection of $R$ about $C$ and the reflection of $S$ about $B$ , respectively. Suppose that points $P$ and $Q$ lie on the lines $BS$ and $RC$ , respectively, such that $PK$ and $QL$ are perpendicular to $BC$ . Prove that the circle with center $P$ and radius $PK$ is tangent to the circumcircle of $RCE$ if and only if the circle with center $Q$ and radius $QL$ is tangent to the circumcircle of $BFS$ .

Proposed by Mehran Talaei