Let $P$ be an interior point of the parallelogram $ABCD$. Prove that $\angle APB+ \angle CPD = 180^\circ$ if and only if $\angle PDC = \angle PBC$.

A polynomial $P(x)$ of degree $n \ge 5$ with integer coefficients has $n$ distinct integer roots, one of which is $0$. Find all integer roots of the polynomial $P(P(x))$.

Let $ABCDEF$ be a convex hexagon such that $AB = BC, CD = DE, EF = FA$. Prove that $\frac{BC}{BE} +\frac{DE}{DA} +\frac{FA}{FC} \ge \frac{3}{2}$ . When does equality occur?

Find all functions $f : N\rightarrow N - \{1\}$ satisfying $f (n)+ f (n+1)= f (n+2) +f (n+3) -168$ for all $n \in N$ .

In a triangle $ABC$, $T$ is the centroid and $ \angle TAB = \angle ACT$. Find the maximum possible value of $sin \angle CAT +sin \angle CBT$.

In a summer camp there are $n$ girls $D_1,D_2, ... ,D_n$ and $2n-1$ boys $C_1,C_2, ...,C_{2n-1}$. The girl $D_i, i = 1,2,... ,n,$ knows only the boys $C_1,C_2, ... ,C_{2i-1}$. Let $A(n, r)$ be the number of different ways in which $r$ girls can dance with $r$ boys forming $r$ pairs, each girl with a boy she knows. Prove that $A(n, r) = \binom{n}{r} \frac{r!}{(n-r)!}.$