2017/18 AHSMC Part 1 Problem 8

Problem 8 of 16

Lan likes red or blue chopsticks, while Dan likes orange or green chopsticks. A drawer in a dark room contains $n$ chopsticks of each colour, where $n > 1$ is an integer. Lan goes into the room to get chopsticks for both her and Dan. What is the smallest number of these $4n$ chopsticks that Lan must pick, to be sure that both people will get a matched (i.e. same colour) pair that they like?

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Solution

If Lan picks up $2n + 2$ chopsticks consisting of $n$ orange, $n$ green, $1$ red and $1$ blue, she does not get a pair she likes. Hence she needs to pick at least $2n + 3$ chopsticks to be sure that she gets a pair she likes. On the other hand, any choice of $2n + 3$ chopsticks will contain at least $3$ red/blue chopsticks and at least $3$ orange/green chopsticks, so both people will get a pair they like. The answer is (B).

University of Alberta

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