JOHN
Lv 7
JOHN asked in Science & MathematicsMathematics · 5 years ago

a, b, c > 0 and abc = 1. Prove that (a + b)(b + c)(c + a) ≥ (a + 1)(b + 1)(c + 1).?

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  • Indica
    Lv 7
    5 years ago
    Favourite answer

    Write p=∑a and q=∑ab

    Inequality becomes (p−a)(p−b)(p−c) ≥ abc+q+p+1

    → p³−p²∑a+p∑ab−abc ≥ abc+q+p+1 → pq−q−p+1 ≥ 2(abc+1)

    So need to prove (p−1)(q−1) ≥ 4

    But AM-GM gives p ≥ 3∛(abc) = 3 and q ≥ 3∛(a²b²c²) = 3 so result is proved.

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