OIM 2019

For each positive integer $n$ , let $s(n)$ be the sum of the squares of the digits of $n$ . For example, $s(15)=1^2+5^2=26$ . Determine all integers $n\geq 1$ such that $s(n)=n$ .

Determine all polynomials $P(x)$ with degree $n\geq 1$ and integer coefficients so that for every real number $x$ the following condition is satisfied $$P(x)=(x-P(0))(x-P(1))(x-P(2))\cdots (x-P(n-1))$$

Let $\Gamma$ be the circumcircle of triangle $ABC$ . The line parallel to $AC$ passing through $B$ meets $\Gamma$ at $D$ ( $D\neq B$ ), and the line parallel to $AB$ passing through $C$ intersects $\Gamma$ to $E$ ( $E\neq C$ ). Lines $AB$ and $CD$ meet at $P$ , and lines $AC$ and $BE$ meet at $Q$ . Let $M$ be the midpoint of $DE$ . Line $AM$ meets $\Gamma$ at $Y$ ( $Y\neq A$ ) and line $PQ$ at $J$ . Line $PQ$ intersects the circumcircle of triangle $BCJ$ at $Z$ ( $Z\neq J$ ). If lines $BQ$ and $CP$ meet each other at $X$ , show that $X$ lies on the line $YZ$ .

Let $ABCD$ be a trapezoid with $AB\parallel CD$ and inscribed in a circumference $\Gamma$ . Let $P$ and $Q$ be two points on segment $AB$ ( $A$ , $P$ , $Q$ , $B$ appear in that order and are distinct) such that $AP=QB$ . Let $E$ and $F$ be the second intersection points of lines $CP$ and $CQ$ with $\Gamma$ , respectively. Lines $AB$ and $EF$ intersect at $G$ . Prove that line $DG$ is tangent to $\Gamma$ .

Don Miguel places a token in one of the $(n+1)^2$ vertices determined by an $n \times n$ board. A move consists of moving the token from the vertex on which it is placed to an adjacent vertex which is at most $\sqrt2$ away, as long as it stays on the board. A path is a sequence of moves such that the token was in each one of the $(n+1)^2$ vertices exactly once. What is the maximum number of diagonal moves (those of length $\sqrt2$ ) that a path can have in total?

Let $a_1, a_2, \dots, a_{2019}$ be positive integers and $P$ a polynomial with integer coefficients such that, for every positive integer $n$ , $$P(n) \text{ divides } a_1^n+a_2^n+\dots+a_{2019}^n.$$ Prove that $P$ is a constant polynomial.