IMO 2010 #3
by Wolstenholme, Nov 4, 2014, 4:03 AM
Find all functions 
such that
![\[
\left(g(m)+n\right)\left(g(n)+m\right)\]](//latex.artofproblemsolving.com/1/f/8/1f87e3d675f33b825ab38662f9eda3ef2e16e66e.png)
is perfect square for all
Solution:
Lemma 1:
and 
Proof: This follows immediately from the fact that the product of two positive integers with difference less than
cannot be a perfect square.
Lemma 2: For any odd prime
, we have that 
for all 
Proof: Assume
for some 
It is clear that there exists a 
such that 
Since 
is a square we must have 
and similarly we find that 
which implies the desired result.
Lemma 3:
for all 
Proof: This is essentially equivalent to the proof of Lemma 2.
The combination of Lemmas
and yields that 
for all 
After some casework using Lemma 1, we find that the only solution is 
where 
such that![\[
\left(g(m)+n\right)\left(g(n)+m\right)\]](http://latex.artofproblemsolving.com/1/f/8/1f87e3d675f33b825ab38662f9eda3ef2e16e66e.png)
is perfect square for all
Solution:
Lemma 1:
and 
Proof: This follows immediately from the fact that the product of two positive integers with difference less than
cannot be a perfect square.Lemma 2: For any odd prime
, we have that 
for all 
Proof: Assume
for some 
It is clear that there exists a 
such that 
Since 
is a square we must have 
and similarly we find that 
which implies the desired result.Lemma 3:
for all 
Proof: This is essentially equivalent to the proof of Lemma 2.
The combination of Lemmas
and yields that 
for all 
After some casework using Lemma 1, we find that the only solution is 
where 
This post has been edited 1 time. Last edited by Wolstenholme, Nov 4, 2014, 3:44 PM
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