An International Mathematical Olympiad level Number Theory problem.

The International Mathematical Olympiad, IMO for short, is known as the toughest competition in the world for high school students. Only a handful of students from each country compete against some of the greatest youthful minds of the world.
In this article, we unravel a problem that was proposed for the IMO in 1992. Do you think you have what it takes to solve it?

At first glance this problem looks fairly simple to solve. It’s a short problem, yet many times short problems prove to be the most difficult. The complexity of the problem comes later on. Naturally, since 25 divides 125, we think of setting a = 5²⁵. Let us now do that:

Now quite a famous factoring trick comes into play, although it is nothing special. That is:

In our case, x=a, y=1, n=5. So we can factor a⁵ -1 as (a-1)(a⁴+a³+a²+a+1), meaning that we can now cancel out the denominator. We are left with just showing that a⁴+a³+a²+a+1 is not prime. Here comes the tricky part. We need to find some way to show that quartic has some factorisation, which is almost never an easy task. So what if we instead tried to write this as…