Let . Prove that

Given a triangle , let be the point which minimizes the sum of squares of distances from the sides of the triangle. Let the projections of on the sides of the triangle . Show that is the barycenter of .

(Jack D’Aurizio)

Let be the set of all positive real numbers. Find all strictly monotone (increasing or decreasing) functions such that there exists a two-variable polynomial with real coefficients satisfying
for all .

Proposed by Navid Safaei, Iran

Given triangle ABC, any line d intersects AB at D, intersects AC at E, intersects BC at F. Let O1,O2,O3 be the centers of the circles circumscribing triangles ADE, BDF, CFE. Prove that the orthocenter of triangle O1O2O3 lies on line d.

Find all integer solutions to

.

Try to avoid Directed angles
Let ABC be an acute triangle inscribed in circle . Let be the midpoint of the arc not containing and define similarly. Show that the orthocenter of is the incenter of .

It is given polygon with sides . His vertices are marked with numbers such that sum of numbers marked by any consecutive vertices is constant and its value is . If we know that is marked with and is marked with , determine with which number is marked

Triangle has two isogonal conjugate points and . The circle intersects circle at , and the circle intersects circle at . Prove that and are isogonal conjugates in triangle .
Note: Circle is the circle with diameter , Circle is the circle with diameter .

In triangle . Points and on the medial perpendicular to are such that . Prove that is the circumcenter of triangle .

Let denote the set of positive real numbers. Find all functions such that
for all

In the right triangle , is the midpoint of the hypotenuse . Point is chosen on the leg so that the line segment meets again at (). Let be the reflection of in . The circles and meet again at (). Find the measure of .