olymmath

ISL 2008 A7

    Prove that for any four positive real numbers a,b,c,da, b, c, d the inequality (a−b)(a−c)a+b+c+(b−c)(b−d)b+c+d+(c−d)(c−a)c+d+a+(d−a)(d−b)d+a+b≥0\frac{(a-b)(a-c)}{a+b+c}+\frac{(b-c)(b-d)}{b+c+d}+\frac{(c-d)(c-a)}{c+d+a}+\frac{(d-a)(d-b)}{d+a+b} \geq 0 holds. Determine all cases of equality.

    Official text Transcription from NuminaMath olympiads, checked word for word against the official paper. Quoted with source.

    Official text: IMO 2008 Shortlist booklet, page 18.

    Solutions

    Official solution Solution 1 by the IMO Problem Selection Committee

    The terms of the source let olymmath link to this solution but not copy it. Read the solution (IMO official site).

    Locator isl/2008/a7 · Tag HRQ777Y5DY · Edit this page on GitHub · Keys: j next, k previous