HMMT 十一月 2012 · 冲刺赛 · 第 9 题

HMMT November 2012 — Guts Round — Problem 9

[ 7 ] How many sets consist of distinct composite numbers that add up to 23? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT NOVEMBER 2012, 10 NOVEMBER 2012 — GUTS ROUND

解析

[ 7 ] Answer: 4 Because 23 is odd, we must have an odd number of odd numbers in our set. Since the smallest odd composite number is 9, we cannot have more than 2 odd numbers, as otherwise the sum would be at least 27. Therefore, the set has exactly one odd number. The only odd composite numbers less than 23 are 9 , 15, and 21. If we include 21, then the rest of the set must include composite numbers that add up to 2, which is impossible. If we include 15, then the rest of the set must include distinct even composite numbers that add up to 8. The only possibility is the set { 8 } . If we include 9, the rest of the set must contain distinct even composite numbers that add to 14. The only possibilities are { 14 } , { 4 , 10 } , and { 6 , 8 } . We have exhausted all cases, so there are a total of 4 sets.