Numbers, sets and sums

A set consists of distinct integers from 1 to 100 inclusive. The set has the property that no two number sum to 125. What is the greatest amount of elements in the set? Source: AMC 10


The answer is 62.

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1 solution

There are 38 38 pairs of integers that sum to 125 , 125, namely

( 25 , 100 ) , ( 26 , 99 ) , ( 27 , 98 ) , . . . . , ( 61 , 64 ) , ( 62 , 63 ) . (25,100), (26,99), (27,98), .... , (61,64), (62,63).

From each of these pairs we can choose only one element to place in the desired set. We can also include all integers from 1 1 to 24 24 as they can't be paired to produce a sum of 125 , 125, and so the desired set has 38 + 24 = 62 38 + 24 = \boxed{62} elements.

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