Hardest problem in International Mathematical Olympiad Ever

Hardest problem in  International Mathematical Olympiad 

The hardest problem ever posed in any International Mathematical Olympiad (IMO) competition is widely regarded as Problem 6 from the 1988 IMO. Before delving into why this problem is considered exceptionally challenging, let's first take a look at the problem itself. 

Here is the infamous Problem 6:

Now, let's delve into why this problem is widely regarded as the most difficult ever presented in the history of the IMO.

The problem demands a deep understanding of functional equations and the properties of integer functions. While it might seem straightforward at first glance, the subtleties involved in proving that \( f \) must be the identity function are far from trivial. Recognizing that the given functional equation is a form of the Cauchy functional equation, which has famously intricate solutions over the real numbers, is key. However, the problem's restriction to integer values adds a unique complexity that requires precise manipulation and logical reasoning. The true challenge and elegance of the problem lie in its apparent simplicity, combined with the profound insight needed to arrive at the correct, yet not immediately obvious, solution.

Solution for problem 6: 

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