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PrintRomanian Mathematical Olympiad
Romania counting and probability
Problem
Iulia and Ştefan shared the 52 playing cards from a deck¹ so each got 26 cards. The cards from 2 to 10 are assigned their own value, the ace is worth 11 points, the jack 12 points, the queen 13 points and the king 14 points. Ştefan noticed that he had no ace in his stack, no 2 and no four cards of the same value. Iulia noticed that from her cards above 11 points, she doesn't have more than two of the same value. Iulia adds up all her points. What is the lowest value she can get? What is the highest?
¹A deck has 52 cards: four with the value 2, four with the value 3, ..., four with the value 10, 4 aces, 4 jacks, 4 queens and 4 kings.
¹A deck has 52 cards: four with the value 2, four with the value 3, ..., four with the value 10, 4 aces, 4 jacks, 4 queens and 4 kings.
Solution
If we denote by the number of cards with the value that Iulia has in her stack, then , , , , . If and represent the highest and the lowest value that Iulia's stack can have then:
Final answer
lowest 169; highest 221
Techniques
Coloring schemes, extremal argumentsGames / greedy algorithms