2024 AMC 10B Problem 5

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 full sum is

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

Flipping a term

t

drops the total by

2t,

so to go negative the flipped terms have to add up to more than

1250.

The greedy move is to flip the biggest odd numbers: flipping the top

k

gives

99 + 97 + \cdots = k(100 - k).

We want

k(100 - k) \gt 1250.

k = 14

it’s only

1204,

but at

k = 15

it jumps to

1275.

15

flips do it. Thus, B is the correct answer.

Tagged: summation · optimization · bounding to limit cases