Find the function $f(x)$ such that \[f(x)^2f\left(\frac{1-x}{x+1}\right) =64x \] for $x\not=0,x\not=1,x\not=-1$.

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.

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 $5$.

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 $CD$.