IGO 2018

There are three rectangles in the following figure. The lengths of some segments are shown. Find the length of the segment $XY$ . Proposed by Hirad Aalipanah

In convex quadrilateral $ABCD$ , the diagonals $AC$ and $BD$ meet at the point $P$ . We know that $\angle DAC = 90^o$ and $2 \angle ADB = \angle ACB$ . If we have $\angle DBC + 2 \angle ADC = 180^o$ prove that $2AP = BP$ .

Proposed by Iman Maghsoudi

Let $\omega_1,\omega_2$ be two circles with centers $O_1$ and $O_2$ , respectively. These two circles intersect each other at points $A$ and $B$ . Line $O_1B$ intersects $\omega_2$ for the second time at point $C$ , and line $O_2A$ intersects $\omega_1$ for the second time at point $D$ . Let $X$ be the second intersection of $AC$ and $\omega_1$ . Also $Y$ is the second intersection point of $BD$ and $\omega_2$ . Prove that $CX = DY$ .

Proposed by Alireza Dadgarnia

We have a polyhedron all faces of which are triangle. Let $P$ be an arbitrary point on one of the edges of this polyhedron such that $P$ is not the midpoint or endpoint of this edge. Assume that $P_0 = P$ . In each step, connect $P_i$ to the centroid of one of the faces containing it. This line meets the perimeter of this face again at point $P_{i+1}$ . Continue this process with $P_{i+1}$ and the other face containing $P_{i+1}$ . Prove that by continuing this process, we cannot pass through all the faces. (The centroid of a triangle is the point of intersection of its medians.)

Proposed by Mahdi Etesamifard - Morteza Saghafian

Suppose that $ABCD$ is a parallelogram such that $\angle DAC = 90^o$ . Let $H$ be the foot of perpendicular from $A$ to $DC$ , also let $P$ be a point along the line $AC$ such that the line $PD$ is tangent to the circumcircle of the triangle $ABD$ . Prove that $\angle PBA = \angle DBH$ .

Proposed by Iman Maghsoudi

Two circles $\omega_1,\omega_2$ intersect each other at points $A,B$ . Let $PQ$ be a common tangent line of these two circles with $P \in \omega_1$ and $Q \in \omega_2$ . An arbitrary point $X$ lies on $\omega_1$ . Line $AX$ intersects $\omega_2$ for the second time at $Y$ . Point $Y'\ne Y$ lies on $\omega_2$ such that $QY = QY'$ . Line $Y'B$ intersects $\omega_1$ for the second time at $X'$ . Prove that $PX = PX'$ .

Proposed by Morteza Saghafian

In acute triangle $ABC, \angle A = 45^o$ . Points $O,H$ are the circumcenter and the orthocenter of $ABC$ , respectively. $D$ is the foot of altitude from $B$ . Point $X$ is the midpoint of arc $AH$ of the circumcircle of triangle $ADH$ that contains $D$ . Prove that $DX = DO$ .

Proposed by Fatemeh Sajadi

Find all possible values of integer $n > 3$ such that there is a convex $n$ -gon in which, each diagonal is the perpendicular bisector of at least one other diagonal.

Proposed by Mahdi Etesamifard

Quadrilateral $ABCD$ is circumscribed around a circle. Diagonals $AC,BD$ are not perpendicular to each other. The angle bisectors of angles between these diagonals, intersect the segments $AB,BC,CD$ and $DA$ at points $K,L,M$ and $N$ . Given that $KLMN$ is cyclic, prove that so is $ABCD$ .

Proposed by Nikolai Beluhov (Bulgaria)

$ABCD$ is a cyclic quadrilateral. A circle passing through $A,B$ is tangent to segment $CD$ at point $E$ . Another circle passing through $C,D$ is tangent to $AB$ at point $F$ . Point $G$ is the intersection point of $AE,DF$ , and point $H$ is the intersection point of $BE$ , $CF$ . Prove that the incenters of triangles $AGF$ , $BHF$ , $CHE$ , $DGE$ lie on a circle.

Proposed by Le Viet An (Vietnam)