PMO 2018

In triangle $ABC$ with $\angle ABC = 60^{\circ}$ and $5AB = 4BC$ , points $D$ and $E$ are the feet of the altitudes from $B$ and $C$ , respectively. $M$ is the midpoint of $BD$ and the circumcircle of triangle $BMC$ meets line $AC$ again at $N$ . Lines $BN$ and $CM$ meet at $P$ . Prove that $\angle EDP = 90^{\circ}$ .

Suppose $a_1, a_2, \ldots$ is a sequence of integers, and $d$ is some integer. For all natural numbers $n$ , \begin{align*}\text{(i)} |a_n| \text{ is prime;} && \text{(ii)} a_{n+2} = a_{n+1} + a_n + d. \end{align*} Show that the sequence is constant.

Let $n$ be a positive integer. An $n \times n$ matrix (a rectangular array of numbers with $n$ rows and $n$ columns) is said to be a platinum matrix if: [list=i] [] the $n^2$ entries are integers from $1$ to $n$ ; [] each row, each column, and the main diagonal (from the upper left corner to the lower right corner) contains each integer from $1$ to $n$ exactly once; and [*] there exists a collection of $n$ entries containing each of the numbers from $1$ to $n$ , such that no two entries lie on the same row or column, and none of which lie on the main diagonal of the matrix. [/list] Determine all values of $n$ for which there exists an $n \times n$ platinum matrix.

Determine all ordered pairs $(x, y)$ of nonnegative integers that satisfy the equation $$3x^2 + 2 \cdot 9^y = x(4^{y+1}-1).$$