IGO2016-P3

Find all positive integers N such that there exists a triangle which can be dissected into N similar quadrilaterals.

IGO 2016-P5

Let the circles ω and ωintersect in points A and B. Tangent to circle ω at A intersects ωin C and tangent to circle ωat A intersects ω in D. Suppose that the internal bisector of ∠CAD intersects ω and ωat E and F, respectively, and the external bisector of ∠CAD intersects ω and ωin X and Y, respectively. Prove that the perpendicular bisector of XY is tangent to the circumcircle of triangle BEF

3rd Iranian Geometry Olympiad-P4

Let ω be the circumcircle of right-angled triangle ABC (∠A = 90◦). Tangent to ω at point A intersects the line BC in point P. Suppose that M is the midpoint of (the smaller) arc AB, and PM intersects ω for the second time in Q. Tangent to ω at point Q intersects AC in K. Prove that ∠PKC = 90◦.

Proposed by Davood Vakili.