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(a + b + c)^2/(a^2+b^2+c^2)+...+ (d+a+b)^5/(d^5+a^5+b^5)<=120

Source: Mediterranean Mathematical Olympiad 2022 P3 MMC

September 21, 2022
algebrainequalities

Problem Statement

Let a,b,c,da, b, c, d be four positive real numbers. Prove that (a+b+c)2a2+b2+c2+(b+c+d)3b3+c3+d3+(c+d+a)4c4+d4+a4+(d+a+b)5d5+a5+b5≤120\frac{(a + b + c)^2}{a^2+b^2+c^2}+\frac{(b + c + d)^3}{b^3+c^3+d^3}+\frac{(c+d+a)^4}{c^4+d^4+a^4}+\frac{(d+a+b)^5}{d^5+a^5+b^5}\le 120
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