IGO 2015

Let ABC be a triangle with $\angle A = 60^o$. The points $M,N,K$ lie on $BC,AC,AB$ respectively such that $BK = KM = MN = NC$. If $AN = 2AK$, find the values of $\angle B$ and $\angle C$.

by Mahdi Etesami Fard

In rectangle $ABCD$, the points $M,N,P,Q$ lie on $AB,BC,CD,DA$ respectively such that the area of triangles $AQM,BMN,CNP,DPQ$ are equal. Prove that the quadrilateral $MNPQ$ is parallelogram.

by Mahdi Etesami Fard

Do there exist $6$ circles in the plane such that every circle passes through centers of exactly $3$ other circles?

by Morteza Saghafian

In acute-angled triangle $ABC$, $BH$ is the altitude of the vertex $B$. The points $D$ and $E$ are midpoints of $AB$ and $AC$ respectively. Suppose that $F$ be the reflection of $H$ with respect to $ED$. Prove that the line $BF$ passes through circumcenter of $ABC$.

by Davood Vakili

In triangle $ABC$, $M,N,K$ are midpoints of sides $BC,AC,AB$, respectively.Construct two semicircles with diameter $AB,AC$ outside of triangle $ABC$. $MK,MN$ intersect with semicircles in $X,Y$. The tangents to semicircles at $X,Y$ intersect at point $Z$. Prove that $AZ \perp BC$. (Mehdi E'tesami Fard)

In triangle $ABC$, we draw the circle with center $A$ and radius $AB$. This circle intersects $AC$ at two points. Also we draw the circle with center $A$ and radius $AC$ and this circle intersects $AB$ at two points. Denote these four points by $A_1, A_2, A_3, A_4$. Find the points $B_1, B_2, B_3, B_4$ and $C_1, C_2, C_3, C_4$ similarly. Suppose that these $12$ points lie on two circles. Prove that the triangle $ABC$ is isosceles.