IT-CESENATICO 2014

Let $ABC$ be a triangle. Let $H$ be the foot of the altitude from $C$ on $AB$ . Suppose that $AH = 3HB$ . Suppose in addition we are given that

(a) $M$ is the midpoint of $AB$ ; (b) $N$ is the midpoint of $AC$ ; (c) $P$ is a point on the opposite side of $B$ with respect to the line $AC$ such that $NP = NC$ and $PC = CB$ .

Prove that $\angle APM = \angle PBA$ .

For any positive integer $n$ , let $D_n$ denote the greatest common divisor of all numbers of the form $a^n + (a + 1)^n + (a + 2)^n$ where $a$ varies among all positive integers.

(a) Prove that for each $n$ , $D_n$ is of the form $3^k$ for some integer $k \ge 0$ . (b) Prove that, for all $k\ge 0$ , there exists an integer $n$ such that $D_n = 3^k$ .

Let $\omega$ be a circle with center $A$ and radius $R$ . On the circumference of $\omega$ four distinct points $B, C, G, H$ are taken in that order in such a way that $G$ lies on the extended $B$ -median of the triangle $ABC$ , and H lies on the extension of altitude of $ABC$ from $B$ . Let $X$ be the intersection of the straight lines $AC$ and $GH$ . Show that the segment $AX$ has length $2R$ .

Prove that there exists a positive integer that can be written, in at least two ways, as a sum of $2014$ -th powers of $2015$ distinct positive integers $x_1 <x_2 <\cdots <x_{2015}$ .

A $(2n + 1) \times (2n + 1)$ grid, with $n> 0$ , is colored in such a way that each of the cell is white or black. A cell is called special if there are at least $n$ other cells of the same color in its row, and at least another $n$ cells of the same color in its column.

(a) Prove that there are at least $2n + 1$ special boxes. (b) Provide an example where there are at most $4n$ special cells. (c) Determine, as a function of $n$ , the minimum possible number of special cells.