1952 AMC 12 Problem 45

Question

If

a

and

b

are two unequal positive numbers, then:

\dfrac{2ab}{a+b}\gt\sqrt{ab}\gt\dfrac{a+b}{2}

\sqrt{ab}\gt\dfrac{2ab}{a+b}\gt\dfrac{a+b}{2}

\dfrac{2ab}{a+b}\gt\dfrac{a+b}{2}\gt\sqrt{ab}

\dfrac{a+b}{2}\gt\dfrac{2ab}{a+b}\gt\sqrt{ab}

\dfrac{a+b}{2}\gt\sqrt{ab}\gt\dfrac{2ab}{a+b}

Recognize the arithmetic, geometric, and harmonic means of

a

and

b

For unequal positive numbers, the mean inequalities are strict

The arithmetic-geometric mean inequality gives

\frac{a+b}{2}\gt\sqrt{ab}

for unequal positive

a,b.

Applying the same inequality to

\frac{1}{a}

and

\frac{1}{b}

gives

\sqrt{ab}\gt\frac{2ab}{a+b}.

Therefore the decreasing order is arithmetic mean, geometric mean, harmonic mean.

Thus, the correct answer is E.

Problem 45 in other years