Problem:
In the following expression, Melanie changed some of the plus signs to minus signs:
When the new expression was evaluated, it was negative. What is the least number of plus signs that Melanie could have changed to minus signs?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
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 numbers of the expression will stay positive and the last will be made negative for the greatest value of that gives a negative value.
Recall that ; that is, the sum of the first odd positive integers is equal to . The expression in the problem statement is the sum of the first 50 odd positive integers, so it equals . Hence must be strictly less than . Because and , at least plus signs must be switched to minus signs for the expression to evaluate to a negative value. Indeed,
The problems on this page are the property of the MAA's American Mathematics Competitions