A natural number is written in each square of an chess board. The allowed move is to add an integer to each of two adjacent numbers in such a way that non-negative numbers are obtained. (Two squares are adjacent if they have a common side.) Find a necessary and sufficient condition for it to be possible for all the numbers to be zero after finitely many operations.

Let be positive integers such that is an integer. Prove that are both the th power of positive integers.

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Epsilon 1.3. Let ABC be a triangle and let D, E, F be the feet of the altitudes, with D on BC, E on CA, and F on AB. Let the parallel through D to EF meet AB at X and AC at Y. Let T be the intersection of EF with BC and let M be the midpoint of side BC. Prove that the points T, M, X, Y are concyclic.

Let and Show that
Where
Let and Show that
Where

Prove that there exists infinitely many positive integers n such that
ϕ(n) | n+1

Given triangle ABC. Outside the triangle, construct rectangles ACDE and BCFG with equal areas. Let M be the midpoint of DF. Prove that CM passes through the center of the circle circumscribing triangle ABC.

Let be positive integers such that is not squarefree. Prove that there exist a positive real , such that and are coprime for any positive integer .

Let and Show that
Where
Where

Let be a quadrilateral inscribed in a circle . The lines and meet at , the lines and meet at , and the diagonals and meet at . Let be the midpoint of the segment , and let be the common point of the segment and the circle . Prove that the circumcircle of the triangle and are tangent to one another.