By Mildorf T.J.
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Example text
Let a, b, c be positive reals such that a + b + c = 1. Prove that √ √ √ √ √ √ ab + c + bc + a + ca + b ≥ 1 + ab + bc + ca 13. (APMO 2005/2) Let a, b, c be positive reals with abc = 8. Prove that a2 (a3 + 1) (b3 + 1) + b2 (b3 + 1) (c3 + 1) + c2 (c3 + 1) (a3 + 1) ≥ 4 3 14. Show that for all positive reals a, b, c, a3 b3 c3 + + ≥a+b+c b2 − bc + c2 c2 − ca + a2 a2 − ab + b2 15. (USAMO 97/5) Prove that for all positive reals a, b, c, a3 1 1 1 1 + 3 + 3 ≤ 3 3 3 + b + abc b + c + abc c + a + abc abc 16.
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.
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.