HMMT 十一月 2024 · 团队赛 · 第 1 题

HMMT November 2024 — Team Round — Problem 1

[20] The integers from 1 to 9 are arranged in a 3 × 3 grid. The rows and columns of the grid correspond to 6 three-digit numbers, reading rows from left to right, and columns from top to bottom. Compute the least possible value of the largest of the 6 numbers. √ √

解析

[20] The integers from 1 to 9 are arranged in a 3 × 3 grid. The rows and columns of the grid correspond to 6 three-digit numbers, reading rows from left to right, and columns from top to bottom. Compute the least possible value of the largest of the 6 numbers. Proposed by: Srinivas Arun Answer: 523 Solution: The 5 cells that make up the top row and left column are all leading digits of the three-digit numbers. Therefore, the largest number has leading digit at least 5, achievable only if 6, 7, 8, and 9 are placed in the bottom right 2 × 2 square. Then, the only three-digit numbers with tens digit less than 6 are the top row and the left column, so unless 5 is in the top left corner, the three-digit number starting with 5 will be at least 560. Now observe 5 is next to two other digits; if they are not 1 or 2 in some order, then either the top row or left column will read at least 530. Thus we can assume 5 is next to 1 or 2. The next-smallest remaining digit is 3, so the three-digit number starting with 52 must be at least 523 . This is achievable as shown below. 5 2 3 1 6 7 4 8 9 √ √