HMMT 二月 2006 · TEAM1 赛 · 第 11 题

HMMT February 2006 — TEAM1 Round — Problem 11

[15] The lottery cards of a certain lottery contain all nine-digit numbers that can be formed with the digits 1, 2 and 3. There is exactly one number on each lottery card. There are only red, yellow and blue lottery cards. Two lottery numbers that differ from each other in all nine digits always appear on cards of different color. Someone draws a red card and a yellow card. The red card has the number 122 222 222 and the yellow card has the number 222 222 222. The first prize goes to the lottery card with the number 123 123 123. What color(s) can it possibly have? Prove your answer.

解析

[15] The lottery cards of a certain lottery contain all nine-digit numbers that can be formed with the digits 1, 2 and 3. There is exactly one number on each lottery card. There are only red, yellow and blue lottery cards. Two lottery numbers that differ from each other in all nine digits always appear on cards of different color. Someone 6 draws a red card and a yellow card. The red card has the number 122 222 222 and the yellow card has the number 222 222 222. The first prize goes to the lottery card with the number 123 123 123. What color(s) can it possibly have? Prove your answer. Answer: The card with the number 123 123 123 is red. Solution: First, it can in fact be red, if, say, cards are colored based on the first digit only (1 = red, 2 = yellow, 3 = blue). We now endeavor to show it must be red. Consider the cards 333 133 133 and 331 331 331: they each differ in all their digits from 122 222 222 and from 222 222 222, so they must both be blue. Now 211 311 311 differs in all its digits from both 122 222 222 and 333 133 133, so it must be yellow. Finally, 123 123 123 differs in all its digits from both 331 331 331 and 211 311 311, so it must be red.