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Prove this inequality

Source: 2019 Jozsef Wildt International Math Competition-W. 34

May 19, 2020
inequalities

Problem Statement

Let aa, bb, cc be positive real numbers and let mm, nn (m≥n)(m \geq n) be positive integers. Prove thatan−1bn−1cm−n−1am+n+bm+n+anbncm−n+bn−1bcn−1am−n−1bm+n+cm+n+bncnam−n+cn−1an−1bm−n−1cm+n+am+n+cnanbm−n≤1abc\frac{a^{n-1}b^{n-1}c^{m-n-1}}{a^{m+n}+b^{m+n}+a^nb^nc^{m-n}}+\frac{b^{n-1}bc^{n-1}a^{m-n-1}}{b^{m+n}+c^{m+n}+b^nc^na^{m-n}}+\frac{c^{n-1}a^{n-1}b^{m-n-1}}{c^{m+n}+a^{m+n}+c^na^nb^{m-n}}\leq \frac{1}{abc}
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