OIM 1987

The sequence $(p_n)$ is defined as follows: $p_1=2$ and for all $n$ greater than or equal to $2$ , $p_n$ is the largest prime divisor of the expression $p_1p_2p_3\ldots p_{n-1}+1$ . Prove that every $p_n$ is different from $5$ .

In a triangle $ABC$ , $M$ and $N$ are the respective midpoints of the sides $AC$ and $AB$ , and $P$ is the point of intersection of $BM$ and $CN$ . Prove that, if it is possible to inscribe a circle in the quadrilateral $AMPN$ , then the triangle $ABC$ is isosceles.

Prove that if $m,n,r$ are positive integers, and: $$1+m+n\sqrt{3}=(2+\sqrt{3})^{2r-1}$$ then $m$ is a perfect square.

Let $r,s,t$ be the roots of the equation $x(x-2)(3x-7)=2$ . Show that $r,s,t$ are real and positive and determine $\arctan r+\arctan s +\arctan t$ .

Let $ABCD$ be a convex quadrilateral and let $P$ and $Q$ be the points on the sides $AD$ and $BC$ respectively such that $\frac{AP}{PD}=\frac{BQ}{QC}=\frac{AB}{CD}$ . Prove that the line $PQ$ forms equal angles with the lines $AB$ and $CD$ .