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