Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 6.2 National olympiad Prove it

Example 4 Given the sides of $\triangle A B C$ are $a, b, c$ , and the semi-perimeter is $p$ , prove: $\sum a^{3}(p-a) \leqslant a b c p$ , equality holds if and only if $\triangle A B C$ is an equilateral triangle.

Solution

Prove $\quad \sum a^{3}(p-a) \leqslant a b c p \Leftrightarrow \sum a^{2}(a-b) (a-c) \geqslant 0$ (Schur's inequality for $r=2$ case) is proved.

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