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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 a2 + b2 + c2 = 6 Find Pmin where
P = a
bc + 2b
ca + 5c
ab
4, For nonnegative reals a, b, c Prove that:
(a + b)2(a + c)2
(b2 −c2)2 + (b + c)2(a + b)2
(c2 −a2)2 + (b + c)2(c + a)2
(a2 −b2)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
a2 + b2 + c2 ≥18
5
6,For positive a, b, c;show that :
(a2 + bc)(b2 + ca)(c2 + ab) (a2 + b2)(b2 + c2)(c2 + a2) + (a −b)(a −c)
b2 + c2 + (b −c)(b −a)
c2 + a2 + (c −a)(c −b)
a2 + b2 ≥1
7,For positive reals a, b, c prove that :
1 + ab + bc + ca
a2 + b2 + c2 ≥ 16abc (a + b)(b + c)(c + a)
8, For nonnegative a, b, c prove that :
(a2 + b2 + c2 −1)2 ≥2(a3b + b3c + c3a −1)
9, Let a, b, c ≥0 satisfy: a2 + b2 + c2 = 2 Find Pmax if
P = (a5 + b5)(b5 + c5)(c5 + a5)
10, a, b, c ≥0. Prove that
(a2 + 5bc)(b2 + 5bc)(c2 + 5ab) ≥27abc(a + b)(b + c)(c + a)
1
11, (Vasile Cirtoaje) a, b, c ≥0. Prove that:
(2a2 + 7bc)(2b2 + 7ca)(2c2 + 7ab) ≥27(ab + bc + ca)3
12, a, b, c ≥0; a + b + c = 3 Prove that
a3 + b3 + c3 + 9abc ≤2[ab(a + b) + bc(b + c) + ca(c + a)]
13, Let a, b, c > 0. Show that:
a2
2a2 + (b + c −a)2 + b2
2b2 + (c + a −b)2 + c2
2c2 + (a + b −c)2 ≤1
14,Leta, b, c ≥0 Show that:
3a2 + 5ab
(b + c)2 + 3b2 + 5bc
(c + a)2 + 3c2 + 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(a2 + b2 + c2)(a −b)(b −c)(c −a)
16, Let a, b, c > 0 satisfy a + b + c = 3. Prove that:
(a3 + b3 + c3)(ab + bc + ca)8 ≤39
17, a, b, c ≥0 Show that
a2
(b + c)2 + b2
(c + a)2 + c2
(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:
(ab3 + bc3 + ca3)(ab + bc + ca) ≤16
19,a, b, c > 0 satisfy abc = 1. Prove that
a √
b2 + 2c + b √
c2 + 2a + c √
a2 + 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:
a2
(b + c)2 + b2
(c + a)2 + c2
(a + b)2 + 1
2 ≥5
4 a2 + b2 + c2
ab + bc + ca
22, For nonnegative a, b, c; show that
3(a4 + b4 + c4) + 7(a2b2 + b2c2 + c2a2) ≥2(a3b + b3c + c3a) + 8(ab3 + bc3 + ca3)
2
23, For a, b, c ≥0 , show that:
a4
(a + b)4 + b4
(b + c)4 + c4
(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 r
a3 + 4 a2 + 4 + 3 r
b3 + 4 b2 + 4 + 3 r
c3 + 4 c2 + 4 ≥3
25, For positive reals a, b, c show that:
5 + 3abc a3 + b3 + c3 ≥4 ab a2 + b2 + bc b2 + c2 + ca c2 + a2
26, For nonnegative a, b, c show that:
a2 + b2 + c2
ab + bc + ca + a(b + c)
b2 + c2 + b(c + a)
c2 + a2 + c(a + b)
a2 + b2 ≥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 a3 + b3 + c3 ≥ 32abc 3(a + b)(b + c)(c + a)
29, For nonnegative reals a, b, c show that:
3 s
(a2 + bc)(b + c)
a(b2 + c2) + 3 s
(b2 + ca)(c + a)
b(c2 + a2) + 3 s
(c2 + ab)(a + b)
c(a2 + b2) ≥3 3√
2
30, For nonnegative a, b, c show that
a √
b2 + bc + c2 + b √
c2 + ca + a2 + c √
a2 + ab + b2 ≥ 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
(a2 + b2 + c2)3 ≥27(a −b)2(b −c)2(c −a)2
33, For nonnegative a, b, c show that
(a2 + b2 + c2)3 ≥2(a −b)2(b −c)2(c −a)2
3
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