Maths Olympiad Prep

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Algebra Difficulty 6.3 National olympiad Prove it

Example 2 Let a,b,c,d>0a, b, c, d>0, and d=max⁡{a,b,c,d}d=\max \{a, b, c, d\}, prove: a(d−b)+b(d−c)+c(d−a)<d2a(d-b)+b(d-c)+c(d-a)<d^{2}.

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Solution

Prove that the right side of equation (4) minus the left side, and rearranging with respect to dd, yields
d2−d(a+b+c)+(ab+bc+ca)d^{2}-d(a+b+c)+(a b+b c+c a)

Recalling the identity
(d−a)(d−b)(d−c)=d3−d2(a+b+c)+d(ab+bc+ca)−abc\begin{aligned} & (d-a)(d-b)(d-c) \\ = & d^{3}-d^{2}(a+b+c)+d(a b+b c+c a)-a b c \end{aligned}

By comparing (5) and (6), we can see that (4) holds.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.