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?

$14$

$15$

$16$

$17$

$18$

Answer: B

Concepts:summation optimization bounding to limit cases

Difficulty rating: 1250

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 $k(100 - k) \gt 1250.$ At $k = 14$ it's only $1204,$ but at $k = 15$ it jumps to $1275.$ So $15$ flips do it. Thus, B is the correct answer.

2024 AMC 10B Problem 5

Answer: B

Difficulty rating: 1250

Solution:

Problem 5 in Other Years