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Topic Of Nhochnhoc Compiled and translated by: S. Mukherjee (Potla) Email : sayanmukherjee1995@gmail.com 1, For positive reals a, b, c prove that: (a + b + c) 3 ≥ 6 √ 3(a − b)(b − c)(c − a) 2, For nonnegative reals a, b, c prove that ab (a + b) 2 + bc (b + c) 2 + ca (c + a) 2 ≤ 1 4 + 4abc (a + b)(b + c)(c + a) 3, For a, b,c > 0 satisfying a 2 + b 2 + c 2 = 6 Find P min where P = a bc + 2b ca + 5c ab 4, For nonnegative reals a, b, c Prove that: (a + b) 2 (a + c) 2 (b 2 − c 2 ) 2 + (b + c) 2 (a + b) 2 (c 2 − a 2 ) 2 + (b + c) 2 (c + a) 2 (a 2 − b 2 ) 2 ≥ 2 5, For positive reals a, b, c prove that a b + c + b c + a + c a + b + 16 5 . ab + bc + ca a 2 + b 2 + c 2 ≥ 18 5 6,For positive a, b, c;show that : (a 2 + bc)(b 2 + ca)(c 2 + ab) (a 2 + b 2 )(b 2 + c 2 )(c 2 + a 2 ) + (a − b)(a − c) b 2 + c 2 + (b − c)(b − a) c 2 + a 2 + (c − a)(c − b) a 2 + b 2 ≥ 1 7,For positive reals a, b, c prove that : 1 + ab + bc + ca a 2 + b 2 + c 2 ≥ 16abc (a + b)(b + c)(c + a) 8, For nonnegative a, b, c prove that : (a 2 + b 2 + c 2 − 1) 2 ≥ 2(a 3 b + b 3 c + c 3 a − 1) 9, Let a, b, c ≥ 0 satisfy: a 2 + b 2 + c 2 = 2 Find P max if P = (a 5 + b 5 )(b 5 + c 5 )(c 5 + a 5 ) 10, a, b, c ≥ 0. Prove that (a 2 + 5bc)(b 2 + 5bc)(c 2 + 5ab) ≥ 27abc(a + b)(b + c)(c + a) 1 11, (Vasile Cirtoaje) a, b, c ≥ 0. Prove that: (2a 2 + 7bc)(2b 2 + 7ca)(2c 2 + 7ab) ≥ 27(ab + bc + ca) 3 12, a, b, c ≥ 0; a + b + c = 3 Prove that a 3 + b 3 + c 3 + 9abc ≤ 2[ab(a + b) + bc(b + c) + ca(c + a)] 13, Let a, b, c > 0. Show that: a 2 2a 2 + (b + c − a) 2 + b 2 2b 2 + (c + a − b) 2 + c 2 2c 2 + (a + b − c) 2 ≤ 1 14,Leta, b, c ≥ 0 Show that: 3a 2 + 5ab (b + c) 2 + 3b 2 + 5bc (c + a) 2 + 3c 2 + 5ca (a + b) 2 ≥ 6 15, For a, b,c ≥ 0 and k ∈ R find the best constant that satisfies (a + b + c) 5 ≥ k(a 2 + b 2 + c 2 )(a − b)(b − c)(c − a) 16, Let a, b, c > 0 satisfy a + b + c = 3. Prove that: (a 3 + b 3 + c 3 )(ab + bc + ca) 8 ≤ 3 9 17, a, b, c ≥ 0 Show that a 2 (b + c) 2 + b 2 (c + a) 2 + c 2 (a + b) 2 + 10abc (a + b)(b + c)(c + a) ≥ 2 18, For a, b,c ≥ 0 such that a + b + c = 3, prove the following inequality: (ab 3 + bc 3 + ca 3 )(ab + bc + ca) ≤ 16 19,a, b, c > 0 satisfy abc = 1. Prove that a √ b 2 + 2c + b √ c 2 + 2a + c √ a 2 + 2b ≥ √ 3 20, For nonnegative a, b, c satisfying ab + bc + ca = 3, prove that 3(a + b + c) + 2( √ a + √ b + √ c) ≥ 15 21, For nonnegative reals a, b, c prove that: a 2 (b + c) 2 + b 2 (c + a) 2 + c 2 (a + b) 2 + 1 2 ≥ 5 4 a 2 + b 2 + c 2 ab + bc + ca 22, For nonnegative a, b, c; show that 3(a 4 + b 4 + c 4 ) + 7(a 2 b 2 + b 2 c 2 + c 2 a 2 ) ≥ 2(a 3 b + b 3 c + c 3 a) + 8(ab 3 + bc 3 + ca 3 ) 2 23, For a, b,c ≥ 0 , show that: a 4 (a + b) 4 + b 4 (b + c) 4 + c 4 (c + a) 4 + 3abc 2(a + b)(b + c)(c + a) ≥ 3 8 24, Let a, b, c ≥ 0 satisfy a + b + c = 3 Prove that: 3 a 3 + 4 a 2 + 4 + 3 b 3 + 4 b 2 + 4 + 3 c 3 + 4 c 2 + 4 ≥ 3 25, For positive reals a, b, c show that: 5 + 3abc a 3 + b 3 + c 3 ≥ 4 ab a 2 + b 2 + bc b 2 + c 2 + ca c 2 + a 2 26, For nonnegative a, b, c show that: a 2 + b 2 + c 2 ab + bc + ca + a(b + c) b 2 + c 2 + b(c + a) c 2 + a 2 + c(a + b) a 2 + b 2 ≥ 4 27,For nonnegative reals a, b, c prove that: (a − b) 2 (a + b) 2 + (b − c) 2 (b + c) 2 + (c − a) 2 (c + a) 2 + 24(ab + bc + ca) (a + b + c) 2 ≤ 8 28, Let a, b, c ≥ 0; show that : 1 + abc a 3 + b 3 + c 3 ≥ 32abc 3(a + b)(b + c)(c + a) 29, For nonnegative reals a, b, c show that: 3 (a 2 + bc)(b + c) a(b 2 + c 2 ) + 3 (b 2 + ca)(c + a) b(c 2 + a 2 ) + 3 (c 2 + ab)(a + b) c(a 2 + b 2 ) ≥ 3 3 √ 2 30, For nonnegative a, b, c show that a √ b 2 + bc + c 2 + b √ c 2 + ca + a 2 + c √ a 2 + ab + b 2 ≥ a + b + c √ ab + bc + ca 31, For nonnegative a, b, c show that (a + b + c) 5 ≥ 25 √ 5(ab + bc + ca)(a − b)(b − c)(c − a) 32,For nonnegative a, b, c show that (a 2 + b 2 + c 2 ) 3 ≥ 27(a − b) 2 (b − c) 2 (c − a) 2 33, For nonnegative a, b, c show that (a 2 + b 2 + c 2 ) 3 ≥ 2(a − b) 2 (b − c) 2 (c − a) 2 3 34,For nonnegative a, b, c show that 3 a 5 (b + c) (b 2 + c 2 )(a 2 + bc) 2 + 3 b 5 (c + a) (c 2 + a 2 )(b 2 + ca) 2 + 3 c 5 (a + b) (a 2 + b 2 )(c 2 + ab) 2 ≥ 3 3 √ 4 35, If A, B, C are three angles of an acute triangle, fing P min where: P = 1 sin n A + 1 sin n B + 1 sin n C + cos m A cos m B cos m C 36, For nonnegative a, b, c show that (a 2 + 5b 2 )(b 2 + 5c 2 )(c 2 + 5a 2 ) ≥ 8abc(a + b + c) 3 37, For nonnegative reals a, b, c prove that: 1 + 8abc (a + b)(b + c)(c + a) ≥ 2(ab + bc + ca) a 2 + b 2 + c 2 38, For nonnegative reals a, b, c prove that: 3 + 8abc (a + b)(b + c)(c + a) ≥ 12(ab + bc + ca) (a + b + c) 2 39,For nonnegative reals a, b, c prove that: (a + b + c)(ab + bc + ca) ≥ √ abc + (a + b)(b + c)(c + a) 2 40, a, b, c ≥ 0 Show that a 2 + b 2 + c 2 ab + bc + ca + 1 2 ≥ a b + c + b c + a + c a + b 41,a, b, c ≥ 0 Show that if a + b + c = 5; we have: 10 + ab 2 + bc 2 + ca 2 ≥ 7 8 (a 2 b + b 2 c + c 2 a) 42, a, b, c ≥ 0 Show that a 2 (a − b) 2 + b 2 (b − c) 2 + c 2 (c − a) 2 ≥ 1 43, a, b, c ≥ 0 Show that if they satisfy a + b + c = 3 we always have: ab c + bc a + ca b + abc ≥ 4 44, a, b, c ≥ 0 Show that 5a 2 + 2bc (b + c) 2 + 5b 2 + 2ca (c + a) 2 + 5c 2 + 2ab (a + b) 2 ≥ 21 4 . a 2 + b 2 + c 2 ab + bc + ca 4 45, a, b, c ≥ 0 Show that 3a 2 + 4bc (b + c) 2 + 3b 2 + 4ca (c + a) 2 + 3c 2 + 4ab (a + b) 2 ≥ 7 4 . (a + b + c) 2 ab + bc + ca 46, a, b, c ≥ 0 ; a + b + c = 2 3 √ 12. Show that: 7 1 + a 3 (1 + b 3 )(1 + c 3 ) ≤ 169 47, For a, b,c > 0 satisfying a + b + c = 3; show that: ab c + bc a + ca b + 9abc 4 ≥ 21 4 48,For a, b,c > 0 satisfying a + b + c = 3, prove that if k = 10+4 √ 6 3 we have: 3(a 2 + b 2 + c 2 ) + abc ≥ 1 + √ 3k 49, For a, b,c > 0 satisfying a + b + c = 3., show that a √ a + b + b √ b + c + c √ c + a ≥ 3 √ 2 50, For a, b,c > 0 satisfying a + b + c = 6, Show that (11 + a 2 )(11 + b 2 )(11 + c 2 ) + 120abc ≥ 4320 51,For a, b,c > 0 satisfying ab + bc + ca = 2, show that ab(4a 2 + b 2 ) + bc(4b 2 + c 2 ) + ca(4c 2 + a 2 ) + 7abc(a + b + c) ≥ 16 52, For a, b,c > 0 , show that : a a 2 + 3bc + b b 2 + 3ca + c c 2 + 3ab ≥ 2(ab + bc + ca) 53,For a, b,c > 0 , prove that a 4a 2 + 5bc + b 4b 2 + 5ca + c 4c 2 + 5ab ≥ (a + b + c) 2 54, For positive real numbers a, b, c show that a √ 4a 2 + 5bc + b √ 4b 2 + 5ca + c √ 4c 2 + 5ab ≤ 1 55, For positive real numbers a, b, c show that a a + √ a 2 + 3bc + b b + √ b 2 + 3ca + c c + √ c 2 + 3ab ≤ 1 56, For positive real numbers a, b, c such that ab + bc + ca = 1; show that 1 √ a 2 + b 2 + 1 √ b 2 + c 2 + 1 √ c 2 + a 2 ≥ 2 + 1 √ 2 5 57, For positive real numbers a, b, c satisfying a + b + c = 2; show that 1 √ a 2 + b 2 + 1 √ b 2 + c 2 + 1 √ c 2 + a 2 ≥ 2 + 1 √ 2 58, For positive real numbers a, b, c such that ab + bc + ca = 3, show that a b 3 + abc + b c 3 + abc + c a 3 + abc ≥ 3 2 59, For positive real numbers a, b, c that satisfy a + b + c = 3, show that a 2 b 3 + abc + b 2 c 3 + abc + c 2 a 3 + abc ≥ 3 2 60, For positive real numbers a, b, c that satisfy a + b + c = 3, show that (1 + a 2 )(1 + b 2 )(1 + c 2 ) ≥ (1 + a)(1 + b)(1 + c) 61, For positive real numbers a, b, c , show that a 3 + b 3 + c 3 abc + 24abc (a + b)(b + c)(c + a) ≥ 4( a b + c + b c + a + c a + b ) 62, For positive real numbers a, b, c , show that a(b + c) b 2 + c 2 + b(c + a) c 2 + a 2 + c(a + b) a 2 + b 2 ≥ (a + b + c) 2 a 2 + b 2 + c 2 63, For positive real numbers a, b, c , show that a + b + c ≥ a(b + c) √ a 2 + 3bc + b(c + a) √ b 2 + 3ca + c(a + b) √ c 2 + 3ab 64, For positive real numbers a, b, c , show that a k + b k + c k ≥ a(b k + c k ) √ a 2 + 3bc + b(c k + a k ) √ b 2 + 3ca + c(a k + b k ) √ c 2 + 3ab 65, For positive real numbers a, b, c , show that a 2 + 4bc + b 2 + 4ca + c 2 + 4ab ≥ 15(ab + bc + ca) 66, For positive real numbers a, b, c , show that a 3 b 3 + abc + b 3 c 3 + abc + c 3 a 3 + abc ≥ 3 √ 2 67, For positive real numbers a, b, c , show that (a + b) 2 (b + c) 2 (c + a) 2 ≥ 64 3 abc(a 2 b + b 2 c + c 2 a) 6 68, For positive real numbers a, b, c , show that 3a 3 + abc b 3 + c 3 + 3b 3 + abc c 3 + a 3 + 3c 3 + abc a 3 + b 3 ≥ 6 69, For positive real numbers a, b, c , show that (a 2 + b 2 + c 2 )(a + b + c) ≥ 3 3abc(a 3 + b 3 + c 3 ) 70, For positive real numbers a, b, c , show that (a + b + c) 2 ab + bc + ca ≥ a(b + c) a 2 + bc + b(c + a) b 2 + ca + c(a + b) c 2 + ab ≥ (a + b + c) 2 a 2 + b 2 + c 2 71, For positive real numbers a, b, c , show that (a + b + c) 2 2(ab + bc + ca) ≥ a 2 a 2 + bc + b 2 b 2 + ca + c 2 c 2 + ab Also show that a b + c + b c + a + c a + b ≥ a 2 a 2 + bc + b 2 b 2 + ca + c 2 c 2 + ab 72, For positive real numbers a, b, c , show that 3(a 2 b + b 2 c + c 2 a)(ab 2 + bc 2 + ca 2 ) ≥ abc(a + b + c) 3 73, For positive real numbers a, b, c , show that a 2 b 2 + b 2 c 2 + c 2 a 2 ≥ (a + b + c) 3 3(ab 2 + bc 2 + ca 2 ) 74, For positive real numbers a, b, c , show that 1 8a 2 + bc + 1 8b 2 + ca + 1 8c 2 + ab ≥ 1 ab + bc + ca 75, For positive reals a, b, c , show that 3(a 2 + b 2 + c 2 ) a + b + c ≥ a 2 − ab + b 2 + b 2 − bc + c 2 + c 2 − ca + a 2 ≥ 3(a 2 + b 2 + c 2 ) 76, For positive real numbers a, b, c , show that a 2 − ab + b 2 + b 2 − bc + c 2 + c 2 − ca + a 2 ≤ 3 77, For positive real numbers a, b, c satisfying a + b + c = 3, show that a 2 b + b 2 c + b 2 c + c 2 a + c 2 a + a 2 b ≤ 3 √ 2 7 78, For positive real numbers a, b, c satisfying a + b + c = 3, show that a √ b + c 2 + b √ c + a 2 + c √ a + b 2 ≥ 3 √ 2 79, For positive reals a, b, c , show that 3(a + b + c) 2(ab + bc + ca) ≥ a a 2 + b 2 + b b 2 + c 2 + c c 2 + a 2 80, For positive reals a, b, c , show that ab √ ab + 2c 2 + bc √ bc + 2a 2 + ca √ ca + 2b 2 ≥ √ ab + bc + ca 81, For a, b,c > 0 satisfying a + b + c = 2 3 √ 12, show that 7 1 + a 3 (1 + b 3 )(1 + c 3 ) ≤ 169 82, For positive reals a, b, c such that a + b + c = 3 , prove that ab c + bc a + ca b + abc ≥ 4 83, For positive reals a, b, c such that a + b + c = 3 , prove that 3(a 2 + b 2 + c 2 ) + abc ≥ 1 + √ 3k where k = 10 + 4 √ 6 3 84, For positive reals a, b, c such that a + b + c = 3 , prove that a √ a + b + b √ b + c + c √ c + a ≥ 3 √ 2 85, (For positive reals a, b, c such that a + b + c = 6 , prove that (11 + a 2 )(11 + b 2 )(11 + c 2 ) + 120abc ≥ 4320 86, For positive reals a, b, c such that ab + bc + ca = 2 , prove that ab(4a 2 + b 2 ) + bc(4b 2 + c 2 ) + ca(4c 2 + a 2 ) + 7abc(a + b + c) ≥ 16 87, For positive reals a, b, c , prove that a a 2 + 3bc + b b 2 + 3ca + c c 2 + 3ab ≥ 2(ab + bc + ca) 88, For positive reals a, b, c , prove that a 4a 2 + 5bc + a a 2 + 3bc + a a 2 + 3bc ≥ (a + b + c) 2 8 89, For positive reals a, b, c , prove that a √ 4a 2 + 5bc + b √ 4b 2 + 5ca + c √ 4c 2 + 5ab ≤ 1 90, For positive reals a, b, c , prove that a a + √ a 2 + 3bc + b b + √ b 2 + 3ca + c c + √ c 2 + 3ab ≤ 1 91, For positive reals a, b, c satisfying ab + bc + ca = 1, show that 1 √ a 2 + b 2 + 1 √ b 2 + c 2 + 1 √ c 2 + a 2 ≥ 2 + 1 √ 2 92, For positive reals a, b, c satisfying a + b + c = 2, show that 1 √ a 2 + b 2 + 1 √ b 2 + c 2 + 1 √ c 2 + a 2 ≥ 2 + 1 √ 2 93, For positive reals a, b, c satisfying a + b + c = 3, show that a b 3 + abc + b c 3 + abc + c a 3 + abc ≥ 3 2 94, For positive reals a, b, c satisfying a + b + c = 3, show that a 2 b 3 + abc + b 2 c 3 + abc + c 2 a 3 + abc ≥ 3 2 95,For positive reals a, b, c satisfying a + b + c = 3, show that (1 + a 2 )(1 + b 2 )(1 + c 2 ) ≥ (1 + a)(1 + b)(1 + c) 96, For positive reals a, b, c show that 1 a 2 + bc ≤ 3 √ 3 2 abc(a + b + c) 97, Given that 1 a + 1 b + 1 c ≥ 9(a 3 + b 3 + c 3 ) (a 2 + b 2 + c 2 ) 2 , Prove that (a 2 + b 2 + c 2 ) 2 (ab + bc + ca) ≥ 9abc(a 3 + b 3 + c 3 ) 98, For a, b,c > 0, show that a a 2 + 3bc + b b 2 + 3ca + c c 2 + 3ab ≥ 2(ab + bc + ca) 9 99, For a, b,c > 0, show that a 4a 2 + 5bc + a a 2 + 3bc + a a 2 + 3bc ≥ (a + b + c) 2 100, For a, b,c > 0, show that a √ 4a 2 + 5bc + b √ 4b 2 + 5ca + c √ 4c 2 + 5ab ≤ 1 101, For a, b,c > 0, show that a a + √ a 2 + 3bc + b b + √ b 2 + 3ca + c c + √ c 2 + 3ab ≤ 1 102, For a, b,c > 0, show that a 3 + b 3 + c 3 abc + 24abc (a + b)(b + c)(c + a) ≥ 4( a b + c + b c + a + c a + b ) 103, For a, b,c > 0, show that a(b + c) b 2 + c 2 + b(c + a) c 2 + a 2 + c(a + b) a 2 + b 2 ≥ (a + b + c) 2 a 2 + b 2 + c 2 104, For a, b,c > 0, show that a + b + c ≥ a(b + c) √ a 2 + 3bc + b(c + a) √ b 2 + 3ca + c(a + b) √ c 2 + 3ab 105, For a, b,c > 0, show that a k + b k + c k ≥ a(b k + c k ) √ a 2 + 3bc + b(c k + a k ) √ b 2 + 3ca + c(a k + b k ) √ c 2 + 3ab 106, For a, b,c > 0, show that a 2 + 4bc + b 2 + 4ca + c 2 + 4ab ≥ 15(ab + bc + ca) 107, For a, b,c > 0, show that a 3 b 3 + abc + b 3 c 3 + abc + c 3 a 3 + abc ≥ 3 √ 2 108, For a, b,c > 0, show that (a + b) 2 (b + c) 2 (c + a) 2 ≥ 64 3 abc(a 2 b + b 2 c + c 2 a) 109, For a, b,c > 0, show that 3a 3 + abc b 3 + c 3 + 3b 3 + abc c 3 + a 3 + 3c 3 + abc a 3 + b 3 ≥ 6 10
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