1989 AIME Problems/Problem 13
Let be a subset of such that no two members of differ by or . What is the largest number of elements can have?
We first show that we can choose at most 5 numbers from such that no two numbers have a difference of or . We take the smallest number to be , which rules out . Now we can take at most one from each of the pairs: , , , . Now, . Because this isn't an exact multiple of , we need to consider some numbers separately.
Notice that . Therefore we can put the last numbers into groups of 11. Now let's examine . If we pick from the first numbers, then we're allowed to pick , , , , . This means we get 10 members from the 20 numbers. Our answer is thus .
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