Problem (IMO 2011) :
Let be a real-valued function defined on the set of real numbers that satisfies
for all real numbers and . Prove that for all .

Proposed by Igor Voronovich, Belarus


Solution :

Let be the assertion

Lemma 1 :

Proof :




Lemma 2 :

Proof :


This gives the lemma directly.

Lemma 3 :

Proof :
Follows directly from Lemma 1 and 2.


Lemma 4 :

Proof :




Lemma 5 :

Proof :



Since , using
But this is obviously not true for

Hence there must be equality that is

Lemma 6 :
Proof :


Lemma 7 :
Proof :
Obvious from lemmas 3 and 7.

Main Proof :
Using lemma 7, the given inequality can be rewritten as
Consider
Thus
But Lemma 6 implies

Hence

QED