# 2010 IMO Problems/Problem 3

## Problem

Find all functions such that is a perfect square for all

*Author: Gabriel Carroll, USA*

## Solution

Suppose such function exist then:

Lemma 1)

Assume for contradiction that

has to be a perfect square

but .

A square cannot be between 2 consecutive squares. Contradiction. Thus,

Lemma 2) (we have show that it can't be 0)

Assume for contradiction, that .

Then there must exist a prime number such that and are in the same residue class modulo .

If where is not divisible by .

If .

Consider an such that

, where is not divisible by

If .

Consider an such that

, where is not divisible by

At least one of , is not divisible by . Hence,

At least one of , is divisible by an odd amount of .

Hence, that number is not a perfect square.

Thus, ,