In triangle $ABC$, $D$ is a point on $AB$ and $E$ is a point on $AC$ such that $BE$ and $CD$ are bisectors of $\angle B$ and $\angle C$ respectively. Let $Q,M$ and $N$ be the feet of perpendiculars from the midpoint $P$ of $DE$ onto $BC,AB$ and $AC$, respectively. Prove that $PQ=PM+PN$.

Let $ x_1, x_2,\ldots , x_n$ be positive real numbers such that $ x_1x_2\cdots x_n = 1$. Prove that \[\sum_{i = 1}^n \frac {1}{n - 1 + x_i}\le 1.\]

Find all odd primes $ p$, if any, such that $ p$ divides $ \sum_{n=1}^{103}n^{p-1}$

Let $(O)$ be a circle, and let $ABP$ be a line segment such that $A,B$ lie on $(O)$ and $P$ is a point outside $(O)$. Let $C$ be a point on $(O)$ such that $PC$ is tangent to $(O)$ and let $D$ be the point on $(O)$ such that $CD$ is a diameter of $(O)$ and intersects $AB$ inside $(O)$. Suppose that the lines $DB$ and $OP$ intersect at $E$. Prove that $AC$ is perpendicular to $CE$.

Find all functions $ f : \mathbb R \rightarrow \mathbb R$ such that $ (x + y)(f(x) - f(y)) = (x -y)f(x + y)$ for all $ x, y\in \mathbb R$

Fifty teams participate in a round robin competition over 50 days. Moreover, all the teams (at least two) that show up in any day must play against each other. Prove that on every pair of consecutive days, there is a team that has to play on those two days.