1. Rosa and Savino play the following game with Neapolitan cards (40 cards numbered from 1 to 10 of 4 different suits): initially, they divide the 40 cards (20 for each), then they take turns placing a card on the table. When some of the cards on the table have values that add up to exactly 15, these cards are removed from the game (if there are multiple ways to get a sum of 15, the player who placed the last card decides which cards with a sum of values equal to 15 to remove). At the end of the game, 2 cards are left in Savino's hand (a 5 and a 3), one card on the table (a 9), and one card in Rosa's hand. What is the value of Rosa's card?
Solution: Rosa has an 8.

The sum of the values of all the cards in the game is $\frac{10 \cdot 11}{2} \cdot 4=220$.

The sum of those eliminated is a multiple of 15 (they are removed in groups with a sum equal to 15).

Let $x$ be the value of Rosa's card, then:

$$
220=15 k+5+3+9+x
$$

so $203-x$ must be a multiple of 15. Since $1 \leq x \leq 10$, the only possibility is $x=8$.