2024 AMC 12B 第 5 题

Question

在下面的表达式中，Melanie 把若干个加号改成了减号：

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

新表达式求值后为负数。Melanie 至少需要把多少个加号改成减号？

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

答案：B

知识点：等差数列最优化

难度评级：1340

小提示：

原和为

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

；把值为

v

的项翻成负号会让总和减少

2v

。

The original sum is

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

flipping a term of value

v

lowers the total by

2v

大提示：

要使结果为负，被翻号的项之和必须超过

1250

；优先翻最大的奇数，最大

k

项之和为

k(100-k)

。

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)

解答：

原表达式是前

50

个正奇数之和，等于

50^2 = 2500

。把值为

v

的项从

+

改为

-

，总和会减少

2v

，所以要使结果为负，翻号项的和必须大于

\dfrac{2500}{2} = 1250

。

为使项数尽量少，应从

99, 97, 95, \ldots

开始翻号。最大的

k

个奇数之和为

k(100 - k)

。当

k = 14

时，

14 \cdot 86 = 1204 \le 1250

；当

k = 15

时，

15 \cdot 85 = 1275 \gt 1250

。所以

15

个足够，而

14

个不够。

所以正确答案是 B。

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.

2024 AMC 12B 第 5 题

答案：B

难度评级：1340

小提示：

大提示：

解答：

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