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.