10.1. Petya wrote ten natural numbers on the board, none of which are equal. It is known that among these ten numbers, three can be chosen that are divisible by 5. It is also known that among the ten numbers written, four can be chosen that are divisible by 4. Can the sum of all the numbers written on the board be less than $75?$

(P. Kozhevnikov)
Answer. It can.

Solution. Example: $1,2,3,4,5,6,8,10,12,20$. In this set, three numbers $(5,10,20)$ are divisible by 5, four numbers $(4,8,12,20)$ are divisible by 4, and the total sum is 71.

Remark. It can be proven (but, of course, this is not required in the problem), that in any example satisfying the problem's conditions, the numbers $1,2,3,4,5,8,10,12$ and 20 must necessarily be present, and instead of the number 6, one can take 7 or 9.

Comment. To receive full marks (7 points for the problem), it is sufficient to have a correct example without specifying which numbers are divisible by 4 and 5, and an explicit calculation of the sum is also not required. Any non-working example is scored 0 points.