51st Ukrainian National Mathematical Olympiad, 4th Round

Question

1-a + 1-b + 1-c 2(ab+bc+ca + 2a^2+b^2+c^2). for non-negative a, b, c with a + b + c = 1.

Accepted Answer

From the Cauchy-Schwartz inequality for the collections

(α, β, γ)

and

(α x, β y, γ z)

we obtain:

(α x + β y + γ z) \leq (α + β + γ) (α x + β y + γ z),

and so

a a + b + b b + c + c c + a \leq a a + b + b b + c + c c + a \leq (a + b + c) (a^{2} + ab + b^{2} + b c + c^{2} + c a) \leq 2 (a^{2} + b^{2} + c^{2}) .

similarly,

b a + b + c b + c + a c + a \leq 2 (a^{2} + b^{2} + c^{2}), c a + b + a b + c + b c + a \leq 2 (ab + b c + c a) .

Adding all three inequalities, we get the required one.