1995 AMC 12 第 27 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
27.
考察如下三角形数阵：两侧依次为 、、、、，内部每个数等于上一行相邻两个数之和。图中给出了第 行到第 行。
设 表示第 行各数之和。 除以 的余数是多少？
Consider the triangular array of numbers with along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows through are shown.
Let denote the sum of the numbers in row What is the remainder when is divided by
答案：E
小提示：
通过统计上一行每个数在下一行中被计入的次数，建立相邻两行行和的关系
Relate a row’s sum to the previous row’s sum by counting how often each old entry contributes
大提示：
解所得递推式，再计算 的幂对 的余数
Solve the resulting recurrence, then compute the power of modulo
解答：
上一行的每个数都会贡献给下一行的两个数，而新出现的两个边界数又额外贡献 。因此 ，且 。所以 由于 ，且 ，可得 。因此 ，正确答案是 E。
Every previous entry contributes to two entries of the next row, and the two new boundary entries contribute an additional Thus with Hence Since and we have Therefore and the correct answer is E.
其他年份的第 27 题
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