4. Ten different natural numbers are such that the product of any 5 of them is even, and the sum of all 10 numbers is odd. What is their smallest possible sum
Answer: 65.

Solution. The product of 5 odd numbers is odd $\Rightarrow$ among the given ten numbers, there are no more than 4 odd numbers. 4 odd numbers are also impossible, since then the sum of all numbers would be even, while according to the condition, it is odd. Therefore, there are a maximum of 3 odd numbers. If there are exactly 3, then there are 7 even numbers. The sum will be minimal if we take the first, smallest numbers. Among the odd numbers, these are 1, 3, 5; among the even numbers, they are $-2, 4, 6, 8, 10, 12, 14$. Their sum $=65$. If there are fewer odd numbers, then the maximum of 5 is subtracted from the sum, and the minimum of 16 is added, meaning the sum will be larger. Therefore, the found set provides the minimum sum.

Criteria. Answer - 1 point. Indicating the set of numbers - another 1 point. Justification that the number of odd numbers is no more than $4-2$ points. Justification that the number of odd numbers is not 4 (and generally any even number) - 1 point. Justification that if there are fewer than 3 odd numbers, the sum will be larger - 1 point. Justification that exactly this set of numbers gives the minimum sum - 1 point.