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Let x, y, z > 0 such that a = x + y, b = y + z, c = z + x. Then our inequality is equivalent to √ (x + y)(y + z) ≥ 2 3 cyc 2 x . cyc From Cauchy-Schwarz inequality, (x + y)(y + z) ≥ 3 3 cyc (y + √ zx) cyc ≥2 √ y+4 cyc √ =2 zx cyc 2 x . g. applied restrictions with homogeneous expressions in the variables. For example, in order to show that a3 + b3 + c3 − 3abc ≥ 0, one may assume, WLOG, that abc = 1 or a + b + c = 1 etc. The reason is explained below. Suppose that abc = k 3 . Let a = ka , b = kb , c = kc .
Let a, b, c be nonnegative reals. Prove that ab + bc + ca ≤ 3 3 (a + b) (b + c) (c + a) . 2. For a, b, c > 0 prove that a b c 3 + + ≥ . 3. Let a, b, c be real numbers. Prove that 2 + (abc)2 + a2 + b2 + c2 ≥ 2(ab + bc + ca). 4. (Michael Rozenberg) Let a, b, c be non-negative numbers such that a + b + c = 3. Prove that √ a 2b + c2 + b 2c + a2 + c 2a + b2 ≤ 3 3. 5. For any acute-angled triangle ABC show that s tan A + tan B + tan C ≥ , r where s and r denote the semi-perimeter and the inraduis, respectively.
Prove that a7 b2 + b7 c2 + c7 a2 ≤ 3. 18. (Samin Riasat) Let x, y, z be positive real numbers. Prove that x y z + + ≥ y z x x+y + 2z y+z + 2x z+x . 19. (Samin Riasat) Let x, y, z be positive real numbers. Prove that xy + (x + y)(y + z) yz + (y + z)(z + x) zx 3 ≤ . 2. 1. 3. a3 + a3 + b3 ≥ 3a2 b. 1. Use ab + ab + cb ≥ √ to prove 3 abc a b c a+b+c + + ≥ √ . 4. 1. 5. Prove and use the following: a2 b2 c2 (a + b + c)(a2 + b2 + c2 ) + + ≥ . b c a ab + bc + ca 2 a a = ab+ca . 2. 5. 5. 6. Solution: Note that √ √ a2 +abc c+ab = a(c+a)(a+b) (b+c)(c+a) .
101 short cuts in math anyone can do by Rockmaker G.