Question

三个不同的一位正整数放在底行格子中。相邻格子的数相加，和放在它们上方的格子中。在第二行继续同样过程，得到顶格中的数。顶格中可能的最大数与最小数之差是多少？

Three different one-digit positive integers are placed in the bottom row of cells. Numbers in adjacent cells are added and the sum is placed in the cell above them. In the second row, continue the same process to obtain a number in the top cell. What is the difference between the largest and smallest numbers possible in the top cell?

$16$

$24$

$25$

$26$

$35$

答案：D

知识点：最优化

难度评级：1610

解答：

如果底行格子中的数依次为 $a$ 、 $b$ 和 $c$ ，那么中间一行就是 $a+b$ 和 $b+c$ 。

因此顶格为 $a+2b+c$ 。要使它最小，把 $b=1$ 放在中间，外侧放 $2,3$ ，得到顶格数 $7$ 。

要使它最大，把 $b=9$ 放在中间，外侧放 $7,8$ ，得到顶格数 $33$ 。所求差为 $33-7=26$ 。

所以正确答案是 D。

If the lower cells contain $a,$ $b,$ and $c,$ the middle row will have $a+b$ and $b+c$ .

This means that the top row will have $a+2b+c$ . To minimize this, put $b=1$ in the middle and $2,3$ in the outer cells. This yields a top number of $7$ .

To maximize it, put $b=9$ in the middle and $7,8$ in the outer cells. This yields a top number of $33$ . The desired difference is $33-7=26$ .

Thus, D is the correct answer.

2006 AMC 8 第 22 题

答案：D

难度评级：1610

解答：

其他年份的第 22 题