OMM 2015

Let $ABC$ be an acuted-angle triangle and let $H$ be it's orthocenter. Let $PQ$ be a segment through $H$ such that $P$ lies on $AB$ and $Q$ lies on $AC$ and such that $\angle PHB= \angle CHQ$ . Finally, in the circumcircle of $\triangle ABC$ , consider $M$ such that $M$ is the mid point of the arc $BC$ that doesn't contain $A$ . Prove that $MP=MQ$ Proposed by Eduardo Velasco/Marco Figueroa

Let $n$ be a positive integer and let $k$ be an integer between $1$ and $n$ inclusive. There is a white board of $n \times n$ . We do the following process. We draw $k$ rectangles with integer sides lenghts and sides parallel to the ones of the $n \times n$ board, and such that each rectangle covers the top-right corner of the $n \times n$ board. Then, the $k$ rectangles are painted of black. This process leaves a white figure in the board.

How many different white figures are possible to do with $k$ rectangles that can't be done with less than $k$ rectangles?

Proposed by David Torres Flores

Let $\mathbb{N} =\{1, 2, 3, ...\}$ be the set of positive integers. Let $f : \mathbb{N} \rightarrow \mathbb{N}$ be a function that gives a positive integer value, to every positive integer. Suppose that $f$ satisfies the following conditions: $f(1)=1$ $f(a+b+ab)=a+b+f(ab)$ Find the value of $f(2015)$ Proposed by Jose Antonio Gomez Ortega

Let $n$ be a positive integer. Mary writes the $n^3$ triples of not necessarily distinct integers, each between $1$ and $n$ inclusive on a board. Afterwards, she finds the greatest (possibly more than one), and erases the rest. For example, in the triple $(1, 3, 4)$ she erases the numbers 1 and 3, and in the triple $(1, 2, 2)$ she erases only the number 1,

Show after finishing this process, the amount of remaining numbers on the board cannot be a perfect square.

Let $n$ be a positive integer and let $d_1, d_2, \dots, d_k$ be its positive divisors. Consider the number $$f(n) = (-1)^{d_1}d_1 + (-1)^{d_2}d_2 + \dots + (-1)^{d_k}d_k$$ Assume $f(n)$ is a power of 2. Show if $m$ is an integer greater than 1, then $m^2$ does not divide $n$ .