HMMT 二月 2004 · 冲刺赛 · 第 11 题

HMMT February 2004 — Guts Round — Problem 11

[7] Find all numbers n with the following property: there is exactly one set of 8 different positive integers whose sum is n .

解析

Find all numbers n with the following property: there is exactly one set of 8 different positive integers whose sum is n . Solution: 36 , 37 The sum of 8 different positive integers is at least 1 + 2 + 3 + · · · + 8 = 36, so we must have n ≥ 36. Now n = 36 satisfies the desired property, since in this case we must have equality — the eight numbers must be 1 , . . . , 8. And if n = 37 the eight numbers must be 1 , 2 , . . . , 7 , 9: if the highest number is 8 then the sum is 36 < n , while if the highest number is more than 9 the sum is > 1 + 2 + · · · + 7 + 9 = 37 = n . So the highest number must be 9, and then the remaining numbers must be 1 , 2 , . . . , 7. Thus n = 37 also has the desired property. However, no other values of n work: if n > 37 then { 1 , 2 , 3 , . . . , 7 , n − 28 } and { 1 , 2 , . . . , 6 , 8 , n − 29 } are both sets of 8 distinct positive integers whose sum is n . So n = 36 , 37 are the only solutions.