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Inequality of four variables

Source: 2017 Korea Winter Program Practice Test 2 #4

August 14, 2019
algebrainequalities

Problem Statement

Let a,b,c,da,b,c,d be the area of four faces of a tetrahedron, satisfying a+b+c+d=1a+b+c+d=1. Show that an+bn+cnn+bn+cn+dnn+cn+dn+ann+dn+an+bnn≤1+2n\sqrt[n]{a^n+b^n+c^n}+\sqrt[n]{b^n+c^n+d^n}+\sqrt[n]{c^n+d^n+a^n}+\sqrt[n]{d^n+a^n+b^n} \le 1+\sqrt[n]{2} holds for all positive integers nn.
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