3. Determine the smallest product of positive numbers $a$ and $b$ that satisfy the equation

$$
20 a b=13 a+14 b
$$
Answer: 1.82.

## Solution.

$$
\begin{gathered}
(20 a b)^{2}=(13 a+14 b)^{2}=(13 a)^{2}+(14 b)^{2}+2 \cdot 13 a \cdot 14 b \geq \\
\geq 2 \cdot 13 a \cdot 14 b+2 \cdot 13 a \cdot 14 b=4 \cdot 13 a \cdot 14 b \Rightarrow \\
20^{2} a b \geq 4 \cdot 13 \cdot 14 \Rightarrow 20^{2} a b \geq 4 \cdot 13 \cdot 14 \Rightarrow a b \geq 1.82 .
\end{gathered}
$$

We used the inequality between the arithmetic mean and the geometric mean

$$
(13 a)^{2}+(14 b)^{2} \geq 2 \cdot 13 a \cdot 14 b
$$

Equality is achieved when $13 a=14 b$. For example, when $a=1.4 ; b=1.3$. Answer: 1.82.

## Grading Criteria.

If the answer is provided without indicating the condition for equality, then 4 points.