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Problem 1352

AIME late

Combinatorics Difficulty 5.7 Prove it Olimpiade Italiana Di Matematica · Italy

Rosa and Savino play the following game with the Neapolitan cards (40 cards numbered from 1 to 10 in 4 different suits): initially the 40 cards are divided (20 for each player), then in turn they place a card on the table. When some of the cards present on the table have values whose sum is exactly 15, these cards are removed from the game (if there are several ways of obtaining sum 15, the player who placed the last card decides which are the cards with sum of values equal to 15 to be removed). At the end of the game, Savino has 2 cards left in hand (a 5 and a 3), there is one card on the table (a 9), and Rosa has one card in hand. What is the value of Rosa's card?

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

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 sum equal to 15).
Denoting by $x$ the value of Rosa's card we have:
$220 = 15k + 5 + 3 + 9 + x$
so $203 - x$ must be a multiple of 15. Since $1 \leq x \leq 10$ , the only possibility is $x = 8$ .

Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty, ordering) added by this project.