with real coefficients such that for all reals 
such that 
we have the following relations![\[ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). \]](http://latex.artofproblemsolving.com/4/4/9/4491a53f9c6f96c18e42eb984b8e83566140bfdb.png)
,
and 
Prove that
be a polynomial with integer coefficients. Suppose there exist infinitely many integer pairs 
such that 
.
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This post has been edited 1 time. Last edited by Amir Hossein, Nov 21, 2017, 9:44 AM
Reason: Added the TeX file.
Reason: Added the TeX file.
.]
and similar and 
follows from AM-GM.
as biggest: 
goes, but you have already goodwritten solutions by link.
![\[\frac{1}{2}\cdot\sqrt{7\cdot\sum{a^2}+2\cdot\sum{ab}}\ge\sum{m_a}\ge \frac{3}{2}\cdot\sqrt{2\cdot\sum{ab}-\sum{a^2}}\]](http://latex.artofproblemsolving.com/6/3/6/636dcb345232fd2157b9aaef858a857889eab704.png)
![\[\sqrt{2\cdot\sum{a^2}+\sqrt{3}\Delta}\ge\sum{m_a}\ge\sqrt{\frac{3}{2}}\cdot\sqrt{\sum{a^2}+2\sqrt{3}\Delta}\]](http://latex.artofproblemsolving.com/c/5/f/c5ff5223d92f57b47771afe125921f9149f13756.png)
)
