Question

在学校最后一天，Wonderful 夫人给班上的学生发果冻豆。她给每个男生的果冻豆数等于班上男生人数；给每个女生的果冻豆数等于班上女生人数。她带了 $400$ 颗果冻豆，发完后剩下六颗。班上男生比女生多两人。她班上有多少名学生？

On the last day of school, Mrs. Wonderful gave jelly beans to her class. She gave each boy as many jelly beans as there were boys in the class. She gave each girl as many jelly beans as there were girls in the class. She brought $400$ jelly beans, and when she finished, she had six jelly beans left. There were two more boys than girls in her class. How many students were in her class?

$26$

$28$

$30$

$32$

$34$

答案：B

知识点：二次方程因式分解

难度评级：1600

解答：

设班上男生人数为 $b$ ，女生人数为 $g$ 。由题意， $b = g + 2$ 。

如果每个男生得到 $b$ 颗果冻豆，那么给所有男生共发出 $b^2$ 颗。同样，给所有女生共发出 $g^2$ 颗。

因此因为 $g$ 不能为负，所以 $g = 13$ 。这意味着 $b = 15$ ，所以 $b + g = 28$ 。 $\begin{gather*} b^2 + g^2 = 400 - 6 \\ (g + 2)^2 + g^2 = 394 \\ 2g^2 + 4g + 4 = 394 \\ g^2 + 2g - 195 = 0 \\ (g + 15)(g - 13) = 0. \end{gather*}$

所以正确答案是 B。

Let $b$ be the number of boys in the class and $g$ be the number of girls. From the problem, we get that $b = g + 2.$

If each boy gets $b$ jelly beans, then Mrs. Wonderful will give out a total of $b^2$ jelly beans to all the boys. Similarly, she will give out $g^2$ jelly beans to all the girls.

Therefore, $\begin{gather*} b^2 + g^2 = 400 - 6 \\ (g + 2)^2 + g^2 = 394 \\ 2g^2 + 4g + 4 = 394 \\ g^2 + 2g - 195 = 0 \\ (g + 15)(g - 13) = 0. \end{gather*}$ Since $g$ cannot be negative, we get that $g = 13.$ This means that $b = 15,$ so $b + g = 28.$

Thus, B is the correct answer.

2009 AMC 8 第 23 题

答案：B

难度评级：1600

解答：

其他年份的第 23 题