Problem 8.2. On the board, 23 signs are drawn - some pluses and some minuses. If any 10 of them are chosen, there will definitely be at least one plus among them. If any 15 of them are chosen, there will definitely be at least one minus among them. How many pluses are there in total?
Answer: 14.

Solution. Since among any 10 signs there is a plus, the number of minuses on the board is no more than 9 (otherwise, we could choose 10 minuses).

Since among any 15 signs there is a minus, the number of pluses on the board is no more than 14 (otherwise, we could choose 15 pluses).

Then the total number of signs on the board is no more than $9+14=23$. Since there are exactly 23 according to the condition, we get that the number of minuses on the board is exactly 9, and the number of pluses is exactly 14.