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36 <= 4*sum(a^3) - sum(a^4) <= 48

Source: IMO Shortlist 2010, Algebra 2

July 17, 2011
inequalitiesalgebrapolynomialIMO ShortlistHi

Problem Statement

Let the real numbers a,b,c,da,b,c,d satisfy the relations a+b+c+d=6a+b+c+d=6 and a2+b2+c2+d2=12.a^2+b^2+c^2+d^2=12. Prove that 36≤4(a3+b3+c3+d3)−(a4+b4+c4+d4)≤48.36 \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
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