IR-TST 2004

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.