HMMT 十一月 2025 · 团队赛 · 第 2 题
HMMT November 2025 — Team Round — Problem 2
题目详情
- [25] Mark writes the squares of several distinct positive integers (in base 10) on a blackboard. Given that each nonzero digit appears exactly once on the blackboard, compute the smallest possible sum of the numbers on the blackboard.
解析
- [25] Mark writes the squares of several distinct positive integers (in base 10) on a blackboard. Given that each nonzero digit appears exactly once on the blackboard, compute the smallest possible sum of the numbers on the blackboard. Proposed by: Marin Hristov Hristov Answer: 855 Solution 1: Note that { 1 , 9 , 25 , 36 , 784 } can be on the blackboard and has sum of elements 855. We will prove this is minimal. The squares less than 855 with a 7 are 576, 676, 729, and 784. We can do cases on which of these appears on the blackboard. 576: The square with a 3 cannot be 36. The next-smallest square with a 3 is 324, so the resulting sum is at least 576 + 324 = 900 > 855. © 2025 HMMT 676: 676 repeats a 6, so it cannot appear. 729: The square with a 5 cannot be 25. The next-smallest square with a 5 is 225, so the resulting sum is at least 729 + 225 = 954 > 855. 784: The square with a 2 must be 25 since it is at most 855 − 784 = 71 (the next one is 121). The square with a 3 must be 36 since it is at most 71 (the next one is 324). As a result, the numbers on the blackboard must add to at least 784 + 25 + 36 + 1 + 9 = 855 . Solution 2: After ruling out 576 and 676 in the above solution, note that no square ends in 2, 3, 7, or 8, so these digits contribute at least 20, 30, 70, and 80, respectively, to the sum. Furthermore, the square that contains 7 must be at least 700, so the minimum possible sum is 700 + 20 + 30 + 80 + 1 + 4 + 5 + 6 + 9 = 855 , achieved by { 1 , 9 , 25 , 36 , 784 } as before.