mathschallenge.net

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?