CMO 2013

Determine all polynomials $P(x)$ with real coefficients such that

$$(x+1) P(x-1)-(x-1) P(x)$$

is a constant polynomial.

The sequence $a_{1}, a_{2}, \ldots, a_{n}$ consists of the numbers $1,2, \ldots, n$ in some order. For which positive integers $n$ is it possible that $0, a_{1}, a_{1}+a_{2}, \ldots, a_{1}+a_{2}+\ldots+a_{n}$ all have different remainders when divided by $n+1$ ?

Let $G$ be the centroid of a right-angled triangle $A B C$ with $\angle B C A=90^{\circ}$. Let $P$ be the point on ray $A G$ such that $\angle C P A=\angle C A B$, and let $Q$ be the point on ray $B G$ such that $\angle C Q B=\angle A B C$. Prove that the circumcircles of triangles $A Q G$ and $B P G$ meet at a point on side $A B$.

Let $n$ be a positive integer. For any positive integer $j$ and positive real number $r$, define

$$f_{j}(r)=\min (j r, n)+\min \left(\frac{j}{r}, n\right), \text { and } g_{j}(r)=\min (\lceil j r\rceil, n)+\min \left(\left\lceil\frac{j}{r}\right\rceil, n\right) \text {, }$$

where $\lceil x\rceil$ denotes the smallest integer greater than or equal to $x$. Prove that

$$\sum_{j=1}^{n} f_{j}(r) \leq n^{2}+n \leq \sum_{j=1}^{n} g_{j}(r)$$

Let $O$ denote the circumcentre of an acute-angled triangle $A B C$. A circle $\Gamma$ passing through vertex $A$ intersects segments $A B$ and $A C$ at points $P$ and $Q$ such that $\angle B O P=\angle A B C$ and $\angle C O Q=\angle A C B$. Prove that the reflection of $B C$ in the line $P Q$ is tangent to $\Gamma$.