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2 digits written on 90 cards

Source: X May Olympiad (Olimpiada de Mayo) 2004 L1 P5

September 22, 2022
combinatorics

Problem Statement

There are 9090 cards and two different digits are written on each one: 0101, 0202, 0303, 0404, 0505, 0606, 0707, 0808, 0909, 1010, 1212, and so on up to 9898. A set of cards is correct if it does not contain any cards whose first digit is the same as the second digit of another card in the set. We call the value of a set of cards the sum of the numbers written on each card. For example, the four cards 0404, 3535, 7878 and 9898 form a correct set and their value is 215215, since04+35+78+98=215 04+35+78+98=215. Find a correct set that has the largest possible value. Explain why it is impossible to achieve a correct set of higher value.
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