2024 AMC 12B Problem 5

Question

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?

14

15

16

17

18

The original sum is

1 + 3 + \cdots + 99 = 50^2 = 2500;

flipping a term of value

v

lowers the total by

2v

To go negative the flipped terms must total more than

1250;

flip the largest odd numbers first, where the largest

k

of them sum to

k(100-k)

The original expression sums the first

50

odd numbers, giving

50^2 = 2500.

Changing a term of value

v

from

+

to

-

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.

So

15

sign changes suffice and

14

do not.

Thus, the correct answer is B.

Problem 5 in other years