MEMO 2016

Let $ABC$ be an acute triangle for which $AB \neq AC$ , and let $O$ be its circumcenter. Line $AO$ meets the circumcircle of $ABC$ again in $D$ , and the line $BC$ in $E$ . The circumcircle of $CDE$ meets the line $CA$ again in $P$ . The lines $PE$ and $AB$ intersect in $Q$ . Line passing through $O$ parallel to the line $PE$ intersects the $A$ -altitude of $ABC$ in $F$ .

Prove that $FP = FQ$ .

Let $ABC$ be a triangle for which $AB \neq AC$ . Points $K$ , $L$ , $M$ are the midpoints of the sides $BC$ , $CA$ , $AB$ . The incircle of $ABC$ with center $I$ is tangent to $BC$ in $D$ . A line passing through the midpoint of $ID$ perpendicular to $IK$ meets the line $LM$ in $P$ .

Prove that $\angle PIA = 90 ^{\circ}$ .

A positive integer $n$ is Mozart if the decimal representation of the sequence $1, 2, \ldots, n$ contains each digit an even number of times.

Prove that:

1. All Mozart numbers are even.
2. There are infinitely many Mozart numbers.

For a positive integer $n$ , the equation $a^2 + b^2 + c^2 + n = abc$ is given in the positive integers.

Prove that:

1. There does not exist a solution $(a, b, c)$ for $n = 2017$ .
2. For $n = 2016$ , $a$ is divisible by $3$ for all solutions $(a, b, c)$ .
3. There are infinitely many solutions $(a, b, c)$ for $n = 2016$ .