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A Source Book in Mathematics: v. 2 by David Eugene Smith

By David Eugene Smith

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USAMO 00/6) Let n ≥ 2 be an integer and S = {1, 2, . . , n}. Show that for all nonnegative reals a1 , a2 , . . , an , b1 , b2 , . . , bn , min{ai aj , bi bj } ≤ i,j∈S min{ai bj , aj bi } i,j∈S 29. (Kiran Kedlaya) Show that for all nonnegative a1 , a2 , . . , an , √ √ a1 + a1 a2 + · · · + n a1 · · · an a1 + a2 a1 + · · · + an ≤ n a1 · ··· n 2 n 30. (Vascile Cartoaje) Prove that for all positive reals a, b, c such that a + b + c = 3, a b c 3 + + ≥ ab + 1 bc + 1 ca + 1 2 31. (Gabriel Dospinescu) Prove that ∀a, b, c, x, y, z ∈ R+ | xy + yz + zx = 3, a(y + z) b(z + x) c(x + y) + + ≥3 b+c c+a a+b 32.

Z2n is such that zi+j ≥ xi yj for all i, j ∈ {1, . . , n}. Let M = max{z2 , z3 , . . , z2n }. Prove that M + z2 + z3 + · · · + z2n 2n 2 ≥ x1 + · · · + xn n y1 + · · · + yn n Reid’s official solution. Let max(x1 , . . , xn ) = max(y1 , . √ . , yn ) = 1. ) We will prove M + z2 + · · · + z2n ≥ x1 + x2 + · · · + xn + y1 + y2 + · · · + yn , after which the desired follows by AM-GM. We will show that the number of terms on the left which are greater than r is at least as large as the number of terms on the right which are greater than r, for all r ≥ 0.

Show that 5+ a b c + + ≥ (1 + a)(1 + b)(1 + c) b c a 7. (Valentin Vornicu13 ) Let a, b, c, x, y, z be arbitrary reals such that a ≥ b ≥ c and either x ≥ y ≥ z or x ≤ y ≤ z. Let f : R → R+ 0 be either monotonic or convex, and let k be a positive integer. Prove that f (x)(a − b)k (a − c)k + f (y)(b − c)k (b − a)k + f (z)(c − a)k (c − b)k ≥ 0 13 This improvement is more widely known than the other one in this packet, and is published in his book, Olimpiada de Matematica... de la provocare la experienta, GIL Publishing House, Zalau, Romania.

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