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Four variable inequality

Source: Russian TST 2015, Day 10 P2 (Group NG), P3 (Groups A & B)

April 21, 2023
algebrainequalities

Problem Statement

Let a,b,c,da,b,c,d be positive real numbers satisfying a2+b2+c2+d2=1a^2+b^2+c^2+d^2=1. Prove that a3+b3+c3+d3+abcd(1a+1b+1c+1d)⩽1.a^3+b^3+c^3+d^3+abcd\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right)\leqslant 1.
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