JBMO 2000

Find all positive integers $n\geq 1$ such that $n^2+3^n$ is the square of an integer.

Bulgaria

A half-circle of diameter $EF$ is placed on the side $BC$ of a triangle $ABC$ and it is tangent to the sides $AB$ and $AC$ in the points $Q$ and $P$ respectively. Prove that the intersection point $K$ between the lines $EP$ and $FQ$ lies on the altitude from $A$ of the triangle $ABC$ .

Albania

At a tennis tournament there were $2n$ boys and $n$ girls participating. Every player played every other player. The boys won $\frac 75$ times as many matches as the girls. It is knowns that there were no draws. Find $n$ .

Serbia