Numbers, sets and sums
Probability
Level
2
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 3 8 pairs of integers that sum to 1 2 5 , namely
( 2 5 , 1 0 0 ) , ( 2 6 , 9 9 ) , ( 2 7 , 9 8 ) , . . . . , ( 6 1 , 6 4 ) , ( 6 2 , 6 3 ) .
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 to 2 4 as they can't be paired to produce a sum of 1 2 5 , and so the desired set has 3 8 + 2 4 = 6 2 elements.