Find Integer Solutions to a/(b+c) + b/(a+c) + c/(a+b) = 4

A Classic Integer Fraction Equation Worth Pausing Over

It’s been a while since we had a good problem to chew on — so here’s one that will make you think. The solution is coming later, but for now, as usual, take a moment to appreciate the unexpected beauty and depth hiding inside something that looks so simple.

The Olympiad Fraction Problem

All you need to do is find integers a, b, and c that satisfy the following equation:

a/(b + c) + b/(a + c) + c/(a + b) = 4

Why This Integer Fraction Equation Is Deceptively Hard

That’s it. Three variables, three fractions, one elegant condition. The kind of problem where the statement fits in a single line, yet the path to a solution is anything but obvious. This is exactly the charm of olympiad fraction problems — mathematics concealing surprising complexity behind a facade of simplicity.