, 2024

Answer (B): To minimize the number of minus signs needed to make the expression negative, minus signs should be chosen for all of the largest numbers. Hence the first $k$ numbers of the expression will stay positive and the last $50-k$ will be made negative for the greatest value of $k$ that gives a negative value.
Recall that $1 + 3 + 5 + \cdots + (2n - 1) = n^2$; that is, the sum of the first $n$ odd positive integers is equal to $n^2$. The expression in the problem statement is the sum of the first 50 odd positive integers, so it equals $50^2 = 2500$. Hence $k^2$ must be strictly less than $\frac{2500}{2} = 1250$. Because $35^2 = 1225$ and $36^2 = 1296$, at least 15 plus signs must be switched to minus signs for the expression to evaluate to a negative value. Indeed,
$$1 + 3 + 5 + \cdots + 69 - 71 - 73 - 75 - \cdots - 99 = 1225 - (2500 - 1225) = -50.$$