Inequalities 1 Classical Theorems 1. Show that for positive reals a, b, c

a 2 b + b 2 c + c 2 a ab2 + bc2 + ca2 ≥ 9a 2 b 2 c 2 . 2. Let a, b, c be positive reals such that abc = 1. Prove that a + b + c ≤ a 2 + b 2 + c 2 . 3. Let P(x) be a polynomial with positive coefficients. Prove that if P 1 x ≥ 1 P(x) holds for x = 1, then it holds for all x > 0. 4. Show that for all positive reals a, b, c, d, 1 a + 1 b + 4 c + 16 d ≥ 64 a + b + c + d . 5. (USAMO 1980/5) Show that for all non-negative reals a, b, c ≤ 1, a b + c + 1 + b c + a + 1 + c a + b + 1 + (1 − a)(1 − b)(1 − c) ≤ 1. 6. (USAMO 1977/5) If a, b, c, d, e are positive reals bounded by p and q with 0 < p ≤ q, prove that (a + b + c + d + e) 1 a + 1 b + 1 c + 1 d + 1 e ≤ 25 + 6 rp q − rq p 2 and determine when equality holds. 7. Let a, b, c, be non-negative reals such that a + b + c = 1. Prove that a 3 + b 3 + c 3 + 6abc ≥ 1 4 . 8. (IMO 1995/2) a, b, c are positive reals with abc = 1. Prove that 1 a 3(b + c) + 1 b 3(c + a) + 1 c 3(a + b) ≥ 3 2 . 9. Let a, b, c be positive reals such that abc = 1. Show that 2 (a + 1)2 + b 2 + 1 + 2 (b + 1)2 + c 2 + 1 + 2 (c +

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TopNewestWhat did you want to talk about? Check out the inequalities wikis for more examples! – Calvin Lin Staff · 2 years, 7 months ago

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