3. Determine all pairs $(a, b)$ of natural numbers such that their difference $a-b$ is the fifth power of some prime number and for which $a-4 \sqrt{b}=b+4 \sqrt{a}$.

The written part of the school round in category C takes place

on Thursday, January 25, 2007

so that it starts in the morning and the contestants have 4 hours of pure time to solve the problems. For each problem, a contestant can earn 6 points, and a successful solver is a student who scores 10 points or more. Allowed aids are writing and drawing supplies, school MF tables, and calculators without a graphic display. These details will be communicated to the students before the start of the competition.

## Solutions to the written part of the school round in category $C$