1986 AMC 12 Problem 17
Attempt Problem 17 of the 1986 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1986 AMC 12 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
17.
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.)
Answer: B
Small hint:
For each color, at most one selected sock can remain unpaired
Big hint:
Use the parity of the total to sharpen the four-unpaired bound, then construct a near-miss
Solution:
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.
Problem 17 in other years
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