Track / Stage 4 / 186 of 340 #446 of 1964

Problem 446

AMC 12 late, AIME early

Algebra Difficulty 4.8 Find the answer

The arithmetic and harmonic means of two real numbers $a$ and $b$ are both 2. What are the values of $a$ and $b$ ?

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Official solution

According to the inequalities between the means, we have

$2=\frac{2}{\frac{1}{a}+\frac{1}{b}} \leq \sqrt{a b} \leq \frac{a+b}{2}=2$

Therefore, there must be equality between the three means, so $a=b$ . On the other hand, $2=\frac{a+b}{2}=\frac{2 a}{2}=a$ . Finally, $a$ and $b$ can only be 2.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.