CMOQR 2015

Find all integer solutions to the equation $7x^2y^2 + 4x^2 = 77y^2 + 1260$ .

A polynomial $f(x)$ with integer coefficients is said to be tri-divisible if $3$ divides $f(k)$ for any integer $k$ . Determine necessary and sufficient conditions for a polynomial to be tri-divisible.

Let $N$ be a 3-digit number with three distinct non-zero digits. We say that $N$ is mediocre if it has the property that when all six 3-digit permutations of $N$ are written down, the average is $N$ . For example, $N = 481$ is mediocre, since it is the average of $\{418, 481, 148, 184, 814, 841\}$ .

Determine the largest mediocre number.

Given an acute-angled triangle $ABC$ whose altitudes from $B$ and $C$ intersect at $H$ , let $P$ be any point on side $BC$ and $X, Y$ be points on $AB, AC$ , respectively, such that $PB = PX$ and $PC = PY$ . Prove that the points $A, H, X, Y$ lie on a common circle.