Suppose that $ p$ is a prime number. Prove that for each $ k$, there exists an $ n$ such that: \[ \left(\begin{array}{c}n\\ \hline p\end{array}\right)=\left(\begin{array}{c}n+k\\ \hline p\end{array}\right)\]

Suppose that $ p$ is a prime number. Prove that the equation $ x^2-py^2=-1$ has a solution if and only if $ p\equiv1\pmod 4$.

Suppose that $ ABCD$ is a convex quadrilateral. Let $ F = AB\cap CD$, $ E = AD\cap BC$ and $ T = AC\cap BD$. Suppose that $ A,B,T,E$ lie on a circle which intersects with $ EF$ at $ P$. Prove that if $ M$ is midpoint of $ AB$, then $ \angle APM = \angle BPT$.

Let $ M,M'$ be two conjugates point in triangle $ ABC$ (in the sense that $ \angle MAB=\angle M'AC,\dots$). Let $ P,Q,R,P',Q',R'$ be foots of perpendiculars from $ M$ and $ M'$ to $ BC,CA,AB$. Let $ E=QR\cap Q'R'$, $ F=RP\cap R'P'$ and $ G=PQ\cap P'Q'$. Prove that the lines $ AG, BF, CE$ are parallel.

This problem is generalization of this one. Suppose $G$ is a graph and $S\subset V(G)$. Suppose we have arbitrarily assign real numbers to each element of $S$. Prove that we can assign numbers to each vertex in $G\backslash S$ that for each $v\in G\backslash S$ number assigned to $v$ is average of its neighbors.

$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$