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Difference between revisions of "2010 IMO Problems/Problem 3"

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Problem

Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that $\left(g(m)+n\right)\left(g(n)+m\right)$ is a perfect square for all $m,n\in\mathbb{N}.$

Author: Gabriel Carroll, USA

Solution

Suppose such function $g$ exist then:

Lemma 1) $g(m) \ne g(m+1)$

Assume for contradiction that $g(m) = g(m+1)$

 has to be a perfect square

but $\left(g(m)+m\right)^2<\left(g(m+1)+m\right)\left(g(m)+m+1\right)<\left(g(m)+m+1\right)^2$ .

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

Lemma 2) $|g(m)-g(m+1)| = 1$ (we have show that it can't be 0)

Assume for contradiction, that $|g(m)-g(m+1)| > 1$ . Then there must exist a prime number $p$ such that $g(m)$ and $g(m+1)$ are in the same residue class modulo $p$ .

If $|g(m)-g(m+1)| = p^aq$ where $q$ is not divisible by $p$ .

If $a=1$ .

Consider an $n$ such that $g(m)+n =p^3$

$g(m+1)+n = p^3 \pm pq =p (r)$ , where $r$ is not divisible by $p$

If $a>1$ .

Consider an $n$ such that $g(m)+n =p$

$g(m+1)+n = p \pm p^aq =p (r)$ , where $r$ is not divisible by $p$

At least one of $g(n)+m$ , $g(n)+m+1$ is not divisible by $p$ . Hence,

At least one of $(g(m+1)+n )(g(n)+m +1)$ , $(g(m)+n )(g(n)+m)$ is divisible by an odd amount of $p$ .

Hence, that number is not a perfect square.

Thus, $g(x)=x+k$ , $k\in\mathbb{N}$

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