HMMT 十一月 2012 · 冲刺赛 · 第 9 题
HMMT November 2012 — Guts Round — Problem 9
题目详情
- [ 7 ] How many sets consist of distinct composite numbers that add up to 23?
HMMT NOVEMBER 2012, 10 NOVEMBER 2012 — GUTS ROUND
英文原题
[ 7 ] How many sets consist of distinct composite numbers that add up to 23?
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HMMT NOVEMBER 2012, 10 NOVEMBER 2012 — GUTS ROUND
解析
英文解析
- [ 7 ]
Answer: 4 Because 23 is odd, we must have an odd number of odd numbers in our set. Since thesmallest odd composite number is 9, we cannot have more than 2 odd numbers, as otherwise the sumwould 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 therest of the set must include distinct even composite numbers that add up to 8. The only possibility isthe set { 8 } . If we include 9, the rest of the set must contain distinct even composite numbers that addto 14. The only possibilities are { 14 } , { 4 , 10 } , and { 6 , 8 } . We have exhausted all cases, so there are atotal of 4 sets.