OMM 1987

Prove that if the sum of two irreducible fractions is an integer then the two fractions have the same denominator.

How many positive divisors does number $20!$ have?

Consider two lines $\ell$ and $\ell '$ and a fixed point $P$ equidistant from these lines. What is the locus of projections $M$ of $P$ on $AB$ , where $A$ is on $\ell$ , $B$ on $\ell '$ , and angle $\angle APB$ is right?

Calculate the product of all positive integers less than $100$ and having exactly three positive divisors. Show that this product is a square.

In a right triangle $ABC$ , M is a point on the hypotenuse $BC$ and $P$ and $Q$ the projections of $M$ on $AB$ and $AC$ respectively. Prove that for no such point $M$ do the triangles $BPM, MQC$ and the rectangle $AQMP$ have the same area.

Prove that for every positive integer n the number $(n^3 -n)(5^{8n+4} +3^{4n+2})$ is a multiple of $3804$ .