Triangle satisfies . Let the incenter of triangle be , which touches at , respectively. Let be the midpoint of . Let the circle centered at passing through intersect at , respecively. Let line meet at , line meet at . Prove that the three lines are concurrent.

Let be a triangle and let be its circumcenter and its incenter.
Let be the radical center of its three mixtilinears and let be the isogonal conjugate of .
Let be the Gergonne point of the triangle .
Prove that line is parallel with line .

Hello everyone! I discovered an interesting geometric property and would like to share it with the community. I'm curious if this is a known result and whether it can be generalized.

Problem Statement:
Let
A
B
C
D
E
K
ABCDEK be a convex hexagon with an incircle centered at
O
O. Prove that:

∠
A
O
B
+
∠
C
O
D
+
∠
E
O
K
=
180
∘
∠AOB+∠COD+∠EOK=180
∘

Determine the largest integer for which the following statement holds: there exists at least one triple of integers such that
and all triples of real numbers, satisfying the equations, are such that are integers.

Marek Maruin, Slovakia

Let be a rational function with complex coefficients whose denominator does not have multiple roots. Let be the complex roots of and be the roots of . Suppose that is a simple root of . Prove that

Let be a tetrahedron with circumcenter . Suppose that the points and are interior points of the edges and , respectively. Let and be the centroids of the triangles , and , respectively. Prove that if the plane is tangent to the sphere then

In a graph, no vertex is connected to all of the others. For any pair of vertices not connected there is a vertex adjacent to both. The sum of the squares of the degrees of vertices is where is the number of vertices. What is the length of the shortest possible cycle in the graph?

Proposed by B. Montágh, Memphis

The intersection of two congruent regular pentagons is a decagon with sides of in this order. Prove that

Show that a regular heptagon can be dissected into a finite number of symmetrical trapezoids, all similar to each other.

Proposed by M. Laczkovich, Budapest

In the plane , given a circle and a point in its interior, not coinciding with the center of . Call a point of space, not lying on , a proper projection center if there exists a plane , not passing through , such that, by projecting the points of from to , the projection of is also a circle, and its center is the projection of . Show that the proper projection centers lie on a circle.

The edges of a complete graph on vertices are coloured in two colours. Prove that the number of cycles formed by four edges of the same colour is more than .

Based on a problem proposed by M. Pálfy

Prove that for any positive integers , , and there exists infinitely many positive integers for which and are not relatively primes.

Proposed by Géza Kós

In acute triangle let and denote the feet of the altitudes from and , respectively. Let line intersect circumcircle at points . Similarly, let line intersect circumcircle at points . Prove that the radical axis of circles and passes through the orthocenter of triangle