2024 AMC 12B Problem 5

Problem 5 of 25EasierAlgebra

In the following expression, Melanie changed some of the plus signs to minus signs:

1 + 3 + 5 + 7 + \cdots + 97 + 99

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?

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Solution

The original expression sums the first

50

odd numbers, giving

50^2 = 2500.

Changing a term of value

v

from

+

-

decreases the total by

2v,

so to make the result negative the flipped terms must total more than

\dfrac{2500}{2} = 1250.

To use as few terms as possible, flip the largest odd numbers

99, 97, 95, \ldots

The largest

k

of them sum to

k(100 - k).

With

k = 14

this is

14 \cdot 86 = 1204 \le 1250,

but with

k = 15

it is

15 \cdot 85 = 1275 \gt 1250.

15

sign changes suffice and

14

do not. Thus, the correct answer is B.