ONE
    Númas
    ← Back to the programme
    OlympiadHardInequalities10–12

    Schur's Inequality

    Desigualdade de Schur

    For non-negative reals a, b, c, prove that a^{3} + b^{3} + c^{3} + abc \ge ab(a+b) + bc(b+c) + ca(c+a) - abc.

    Rearrange and factor: this is equivalent to Schur's inequality for t=1.

    Solution

    Step 1 of 13

    1. 1.Schur's inequality for t = 1 states: a(a-b)(a-c) + b(b-a)(b-c) + c(c-a)(c-b) \ge 0.