RS-SMO 2018

Let $\triangle ABC$ be a triangle with incenter $I$ . Points $P$ and $Q$ are chosen on segmets $BI$ and $CI$ such that $2\angle PAQ=\angle BAC$ . If $D$ is the touch point of incircle and side $BC$ prove that $\angle PDQ=90$ .

Let $n>1$ be an integer. Call a number beautiful if its square leaves an odd remainder upon divison by $n$ . Prove that the number of consecutive beautiful numbers is less or equal to $1+\lfloor \sqrt{3n} \rfloor$ .

Prove that there exists a uniqe $P(x)$ polynomial with real coefficients such that $xy-x-y|(x+y)^{1000}-P(x)-P(y)$ for all real $x,y$ .

For each positive integer $k$ , let $n_k$ be the smallest positive integer such that there exists a finite set $A$ of integers satisfy the following properties:

• For every $a\in A$ , there exists $x,y\in A$ (not necessary distinct) that $$n_k\mid a-x-y$$ [/*]
• There's no subset $B$ of $A$ that $|B|\leq k$ and $$n_k\mid \sum_{b\in B}{b}.$$

Show that for all positive integers $k\geq 3$ , we've $$n_k<\Big( \frac{13}{8}\Big)^{k+2}.$$