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Two inequalities from Russia 2016

Source: Russia national 2016

May 1, 2016
inequalitiesalgebra

Problem Statement

All russian olympiad 2016,Day 2 ,grade 9,P8 : Let a,b,c,da, b, c, d be are positive numbers such that a+b+c+d=3a+b+c+d=3 .Prove that1a2+1b2+1c2+1d2≤1a2b2c2d2\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{d^2}\le\frac{1}{a^2b^2c^2d^2} All russian olympiad 2016,Day 2,grade 11,P7 : Let a,b,c,da, b, c, d be are positive numbers such that a+b+c+d=3a+b+c+d=3 .Prove that 1a3+1b3+1c3+1d3≤1a3b3c3d3\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}+\frac{1}{d^3}\le\frac{1}{a^3b^3c^3d^3} Russia national 2016
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