AT-OEMO-FED1 2017

Determine all polynomials $P(x) \in \mathbb R[x]$ satisfying the following two conditions : (a) $P(2017) = 2016$ and (b) $(P(x) + 1)^2 = P(x^2 + 1)$ for all real numbers $x$ .

proposed by Walther Janous

Let $ABCDE$ be a regular pentagon with center $M$ . A point $P$ (different from $M$ ) is chosen on the line segment $MD$ . The circumcircle of $ABP$ intersects the line segment $AE$ in $A$ and $Q$ and the line through $P$ perpendicular to $CD$ in $P$ and $R$ . Prove that $AR$ and $QR$ have same length.

proposed by Stephan Wagner

Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. At the beginning of a turn there are n ≥ 1 marbles on the table, then the player whose turn is removes k marbles, where k ≥ 1 either is an even number with $k \le \frac{n}{2}$ or an odd number with $\frac{n}{2}\le k \le n$ . A player wins the game if she removes the last marble from the table. Determine the smallest number $N\ge100000$ which Berta has wining strategy.

proposed by Gerhard Woeginger

Find all pairs $(a,b)$ of non-negative integers such that: $$2017^a=b^6-32b+1$$ proposed by Walther Janous