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Problem

Prove that (a + b)/2 greater than or equal radical(ab), where a, b are non-negative real numbers.


Solution

Without loss of generality, assume that a less than or equal b; let b = a + k, where k greater than or equal 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 minus k2/4.

Clearly (a + k/2)2 greater than or equal (a + k/2)2 minus k2/4 = ab; that is, ((a + b)/2)2 greater than or equal ab.

Hence (a + b)/2 greater than or equal radical(ab).

Alternatively we begin with the observation that (radicala minus b)2 greater than or equal 0.

Expanding we get, a + b minus 2radical(ab) greater than or equal 0, which leads to (a + b)/2 greater than or equal radical(ab).

Prove that (a + b + c)/3 greater than or equal 3radical(abc).
Can you generalise?

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

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