Given two odd natural numbers prove that for each there exists such that either or is multiple of

Given a convex quadrilateral such that . The lines and intersect at a point and the line passing through which is parallel to , intersects at . Prove that

In acute-angled triangle (), point is the orthocenter and point is the midpoint of the segment . The median intersects the circumcircle of triangle at . The line intersects the perpendicular bisector of at and the circumcircle of the triangle again at . Point lies on circle , passing through and , such that is a trapezoid (). Prove that and meet on .


Proposed by Alireza Dadgarnia

Let
For each , define
Find all functions such that, whenever both and ,

Let be a finite set of points in the plane such that no three of them are on a line. For each convex polygon whose vertices are in , let be the number of vertices of , and let be the number of points of which are outside . A line segment, a point, and the empty set are considered as convex polygons of , , and vertices respectively. Prove that for every real number where the sum is taken over all convex polygons with vertices in .

Alternative formulation:

Let be a finite point set in the plane and no three points are collinear. A subset of will be called round if its elements is the set of vertices of a convex gon For each round subset let be the number of points from which are exterior from the convex gon Subsets with and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset of construct the polynomial

Show that the sum of polynomials for all round subsets is exactly the polynomial

Proposed by Federico Ardila, Colombia

Let be an odd prime. Define over the positive integers as follows:

Let be chosen such that there exists an ordered pair of positive integers where such that . Prove that there exists at least three integers such that and is a perfect square.

Let be a positive integer. Aslı and Zehra are playing a game on an grid. Initially, stones are placed on some of the unit squares of this grid.

On each move (starting with Aslı), Aslı chooses a row or a column that contains at least two squares with different numbers of stones, and Zehra redistributes the stones in that row or column so that after redistribution, the difference in the number of stones between any two squares in that row or column is at most one. Furthermore, this move must change the number of stones in at least one square.

For which values of , regardless of the initial placement of the stones, can Aslı guarantee that every square ends up with the same number of stones?

Find all positive integers for which all positive divisors of can be put into the cells of a rectangular table under the following constraints:
[list]
[*]each cell contains a distinct divisor;
[*]the sums of all rows are equal; and
[*]the sums of all columns are equal.
[/list]

Let be triangle with incenter . A point in the interior of the triangle satisfies Show that , and that equality holds if and only if .

Let be a positive integer. Define a chameleon to be any sequence of letters, with exactly occurrences of each of the letters and . Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon , there exists a chameleon such that cannot be changed to using fewer than swaps.