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

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

解答：

其他年份的第 5 题