IMO Shortlist 2010 problem A2


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June 23, 2013
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Let the real numbers a,b,c,d satisfy the relations a+b+c+d=6 and a^2+b^2+c^2+d^2=12. Prove that
36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.

Proposed by Nazar Serdyuk, Ukraine
Source: Međunarodna matematička olimpijada, shortlist 2010