## Mean Proof

#### Problem

Prove that (a + b)/2 (ab), where a, b are non-negative real numbers.

#### Solution

Without loss of generality, assume that a b; let b = a + k, where k 0.

Therefore, (a + b)/2 = (a + a + k)/2 = (2a + k)/2 = a + k/2.

Also, ab = a(a + k) = a2 + ak = (a + k/2)2 k2/4.

Clearly (a + k/2)2 (a + k/2)2 k2/4 = ab; that is, ((a + b)/2)2 ab.

Hence (a + b)/2 (ab).

Alternatively we begin with the observation that (a b)2 0.

Expanding we get, a + b 2(ab) 0, which leads to (a + b)/2 (ab).

Prove that (a + b + c)/3 3(abc).
Can you generalise?

Problem ID: 230 (10 Jul 2005)     Difficulty: 3 Star

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