IGO 2016

Ali wants to move from point $A$ to point $B$ . He cannot walk inside the black areas but he is free to move in any direction inside the white areas (not only the grid lines but the whole plane). Help Ali to find the shortest path between $A$ and $B$ . Only draw the path and write its length. by Morteza Saghafian

Let $\omega$ be the circumcircle of triangle $ABC$ with $AC > AB$ . Let $X$ be a point on $AC$ and $Y$ be a point on the circle $\omega$ , such that $CX = CY = AB$ . (The points $A$ and $Y$ lie on different sides of the line $BC$ ). The line $XY$ intersects $\omega$ for the second time in point $P$ . Show that $PB = PC$ .

by Iman Maghsoudi

Suppose that $ABCD$ is a convex quadrilateral with no parallel sides. Make a parallelogram on each two consecutive sides. Show that among these $4$ new points, there is only one point inside the quadrilateral $ABCD$ .

by Morteza Saghafian

In a right-angled triangle $ABC$ ( $\angle A = 90^o$ ), the perpendicular bisector of $BC$ intersects the line $AC$ in $K$ and the perpendicular bisector of $BK$ intersects the line $AB$ in $L$ . If the line $CL$ be the internal bisector of angle $C$ , find all possible values for angles $B$ and $C$ .

by Mahdi Etesami Fard

Let $ABCD$ be a convex quadrilateral with these properties: $\angle ADC = 135^o$ and $\angle ADB - \angle ABD = 2\angle DAB = 4\angle CBD$ . If $BC = \sqrt2 CD$ , prove that $AB = BC + AD$ .

by Mahdi Etesami Fard

Let the circles $\omega$ and $\omega^ \prime$ intersect in $A$ and $B$ . Tangent to circle $\omega$ at $A$ intersects $\omega^ \prime$ in $C$ and tangent to circle $\omega^ \prime$ at $A$ intersects $\omega$ in $D$ . Suppose that $CD$ intersects $\omega$ and $\omega^ \prime$ in $E$ and $F$ , respectively (assume that $E$ is between $F$ and $C$ ). The perpendicular to $AC$ from $E$ intersects $\omega^ \prime$ in point $P$ and perpendicular to $AD$ from $F$ intersects $\omega$ in point $Q$ (The points $A, P$ and $Q$ lie on the same side of the line $CD$ ). Prove that the points $A, P$ and $Q$ are collinear. Proposed by Mahdi Etesami Fard

In acute-angled triangle $ABC$ , altitude of $A$ meets $BC$ at $D$ , and $M$ is midpoint of $AC$ . Suppose that $X$ is a point such that $\measuredangle AXB = \measuredangle DXM =90^\circ$ (assume that $X$ and $C$ lie on opposite sides of the line $BM$ ). Show that $\measuredangle XMB = 2\measuredangle MBC$ .Proposed by Davood Vakili

In a convex qualrilateral $ABCD$ , let $P$ be the intersection point of $AD$ and $BC$ . Suppose that $I_1$ and $I_2$ are the incenters of triangles $PAB$ and $PDC$ ,respectively. Let $O$ be the circumcenter of $PAB$ , and $H$ the orthocenter of $PDC$ . Show that the circumcircles of triangles $AI_1B$ and $DHC$ are tangent together if and only if the circumcircles of triangles $AOB$ and $DI_2C$ are tangent together. Proposed by Hooman Fattahimoghaddam

In a convex quadrilateral $ABCD$ , the lines $AB$ and $CD$ meet at point $E$ and the lines $AD$ and $BC$ meet at point $F$ . Let $P$ be the intersection point of diagonals $AC$ and $BD$ . Suppose that $\omega_1$ is a circle passing through $D$ and tangent to $AC$ at $P$ . Also suppose that $\omega_2$ is a circle passing through $C$ and tangent to $BD$ at $P$ . Let $X$ be the intersection point of $\omega_1$ and $AD$ , and $Y$ be the intersection point of $\omega_2$ and $BC$ . Suppose that the circles $\omega_1$ and $\omega_2$ intersect each other in $Q$ for the second time. Prove that the perpendicular from $P$ to the line $EF$ passes through the circumcenter of triangle $XQY$ . Proposed by Iman Maghsoudi

Do there exist six points $X_1,X_2,Y_1, Y_2,Z_1,Z_2$ in the plane such that all of the triangles $X_iY_jZ_k$ are similar for $1\leq i, j, k \leq 2$ ? Proposed by Morteza Saghafian