Let be a semicircle with diameter and midpoint . Let be a point on different from and .

The circle touches in a point , the segment in a point , and additionally the segment . The circle touches in a point and additionally the segments and .

Show that the lines and are perpendicular.

A quadrilateral with no parallel sides is inscribed in a circle . Circles are inscribed in triangles , respectively. Common external tangents are drawn between and , and , and , and and , not containing any sides of quadrilateral . A quadrilateral whose consecutive sides lie on these four lines is inscribed in a circle . Prove that the lines joining the centers of and , and , and the centers of and all intersect at one point.

Suppose that , , , are integers such that .
Prove that there exist two permutations and of such that for each integer with , we have


Proposed by Ricky Liu & Zuming Feng, USA

Let be an odd prime and be integers so that the integers are divisible by .
Prove that divides each of .

Proposed by Navilarekallu Tejaswi

Prove that

Let be real numbers such that Prove that Where

Let be a triangle with circumcircle . Let and respectively denote the midpoints of the arcs and that do not contain the third vertex. Let denote the midpoint of arc (the arc including ). Let be the incenter of . Let be the circle that is tangent to and internally tangent to at , and let be the circle that is tangent to and internally tangent to at . Show that the line , and the lines through the intersections of and , meet on .

Ali is hosting a large party. Together with his friends, people are seated around a circular table in a fixed order. Ali places apples for serving directly in front of himself and wants to distribute them among everyone. Since Ali and his friends dislike eating alone and won't start unless everyone receives an apple at the same time, in each step, each person who has at least one apple passes one apple to the first person to their right who doesn't have an apple (in the clockwise direction).

Find all values of such that after some number of steps, the situation reaches a point where each person has exactly one apple.

Find all positive integers n>2 such that sum of n and any of its prime divisors is a perfect square.

Let be an arbitrary triangle and is the circumcenter of .Points lie on ,respectively such that the reflection of WRT is tangent to circumcircle of .Prove that the circumcircle of triangle is tangent to circumcircle of triangle .

The school A has m1 boys and m2 girls, and ,the school B has n1 boys and n2 girls. Each school is represented by one team formed by p students,boys and girls. If f(k) is the number of cases for which,the twice schools has,togheter k girls, fund f(k) and the valute of k, for which f(k) is maximum.

Let be an odd prime number. Let be the sequence defined as follows: , and for , is the smallest integer greater than such that in there are no arithmetic progressions of length . We say that a positive integer is a ghost if it doesn’t appear in .
What is the smallest ghost that is not a multiple of ?

Proposed by Guerrero