6. From the set $\{1,2, \ldots, 100\}$ we choose 50 distinct numbers, whose sum is 3000. At least how many even numbers have we chosen?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6.
6. The answer is (E). The sum of all positive odd numbers less than 100 is 2500. Therefore, in any subset $X$ of $\{1, \ldots, 100\}$ whose elements sum to 3000, the contribution of the even numbers in $X$ to the sum must be at least 500. Five even numbers are not sufficient (at most one of them can be equal to 100), while six are sufficient (it is enough to take the six largest even numbers). On the other hand, it is easy to construct a set $X$ with the desired properties and with exactly 6 even numbers: for example, take all the odd numbers except for the 6 numbers 1, 3, $5, 7, 9, 45$ and all the even numbers $90, 92, 94, 96, 98, 100$.