5. The smallest set. (From 6th grade, 2 points) In a numerical set, the median is 3, the arithmetic mean is 5, and the only mode of the set is 6. What is the smallest number of numbers that can be in a set with the given properties?
Solution. It is clear that the number 6 appears at least twice in the set, and in addition, there are at least two more numbers.

If the set contains exactly four numbers $a, b, 6, 6$, then we can assume that $a \leq b \leq 3$ and, moreover, the sum of all numbers is 20, so $a+b=20-6-6=8$. Contradiction. The set cannot contain exactly four numbers.

If the set contains exactly five numbers, then the set has the form $a, b, 3, 6, 6$, where $a \leq b \leq 3$ and $a+b=25-6-6-3=10$. Contradiction. The set cannot consist of exactly five numbers.

A set of six numbers with the specified properties exists. For example, $-2, -1, 0, 6, 6, 21$.

## Answer: 6.