1986 AMC 12 第 17 题
先试着解答 1986 AMC 12 第 17 题，然后核对你的答案与精心整理的解答，解答来自 LIVE by Po-Shen Loh。你也可以参加完整限时模拟考试、查看全部 1986 AMC 12 解答，或核对答案。
所有题目均经美国数学协会（MAA）官方合法授权使用。
17.
一间暗室里的抽屉中有 只红袜、 只绿袜、 只蓝袜和 只黑袜。一个孩子从抽屉中逐只取出袜子，但看不到取出的袜子是什么颜色。至少必须取出多少只袜子，才能保证其中至少有 双？（一双袜子是两只颜色相同的袜子。每只袜子至多计入一双。）
A drawer in a darkened room contains red socks, green socks, blue socks and black socks. A youngster selects socks one at a time from the drawer but is unable to see the color of the socks drawn. What is the smallest number of socks that must be selected to guarantee that the selection contains at least pairs? (A pair of socks is two socks of the same color. No sock may be counted in more than one pair.)
答案：B
小提示：
对每种颜色，所取袜子中至多有一只无法配对
For each color, at most one selected sock can remain unpaired
大提示：
利用袜子总数的奇偶性改进“四只未配对”的界，再构造一个只差一点的反例
Use the parity of the total to sharpen the four-unpaired bound, then construct a near-miss
解答：
取出 只袜子时，数量为奇数的颜色种数也必须为奇数，所以至多为 。因此，至多有 只袜子未配对，余下至少 只袜子可组成 双。但是，取出 只还不够：若四种颜色的数量分别为 ，则只能组成 双。因此，最小数量为 。
所以正确答案是 B。
With selected socks, the number of colors having an odd count must itself be odd, so it is at most Thus at most socks are unpaired, leaving at least socks in pairs. But socks do not suffice: color counts produce only pairs. Therefore the minimum is
Thus the correct answer is B.
其他年份的第 17 题
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