DE-MO 2011

Prove for each non-negative integer $n$ and real number $x$ the inequality $$\sin{x} \cdot(n \sin{x}-\sin{nx}) \geq 0$$

The price for sending a packet (a rectangular cuboid) is directly proportional to the sum of its length, width, and height. Is it possible to reduce the cost of sending a packet by putting it into a cheaper packet?

Let $ABC$ be an acute triangle and $D$ the foot of the altitude from $A$ onto $BC$ . A semicircle with diameter $BC$ intersects segments $AB,AC$ and $AD$ in the points $F,E$ resp. $X$ . The circumcircles of the triangles $DEX$ and $DXF$ intersect $BC$ in $L$ resp. $N$ other than $D$ . Prove $BN=LC$ .

There are two points $A$ and $B$ in the plane. a) Determine the set $M$ of all points $C$ in the plane for which $|AC|^2 +|BC|^2 = 2\cdot|AB|^2.$ b) Decide whether there is a point $C\in M$ such that $\angle ACB$ is maximal and if so, determine this angle.

Let $p>2$ be a prime. Define a sequence $(Q_{n}(x))$ of polynomials such that $Q_{0}(x)=1, Q_{1}(x)=x$ and $Q_{n+1}(x) =xQ_{n}(x) + nQ_{n-1}(x)$ for $n\geq 1.$ Prove that $Q_{p}(x)-x^p$ is divisible by $p$ for all integers $x.$