Problem:
From the set we choose 50 distinct numbers, whose sum is 3000. At minimum, how many even numbers have we chosen?
Pick one
Solution
Solution:
The answer is (E). The sum of all positive odd numbers less than 100 is 2500. Therefore in every subset of whose elements sum to 3000, the contribution of the even numbers of to the sum must be at least equal to 500. Five even numbers are not enough (at most one of them can be equal to 100), while six are enough (it suffices to take the six largest even numbers). On the other hand, it is easy to construct a set with the required properties and with exactly 6 even numbers: it suffices to take, for example, all the odd numbers with the exception of the 6 numbers 1, 3, 5, 7, 9, 45 and all the even numbers 90, 92, 94, 96, 98, 100.
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