Question

一本书的页码从 $1$ 编到 $n$ 。将所有页码相加时，误将其中一个页码加了两次，得到错误的和 $1986$ 。被重复相加的页码是多少？

The pages of a book are numbered $1$ through $n.$ When the page numbers of the book were added, one of the page numbers was mistakenly added twice, resulting in an incorrect sum of $1986.$ What was the number of the page that was added twice?

答案

答案：33

知识点：极限情形界定三角形数

难度评级：1490

小提示：

将 $1986$ 与相邻的三角数比较

Compare $1986$ with consecutive triangular numbers

大提示：

超出 $1+2+\cdots+n$ 的部分就是被重复计算的页码

The excess over $1+2+\cdots+n$ is the repeated page number

解答：

$1986$ 附近的相邻三角数为 $\begin{aligned} \frac{62\cdot63}{2}&=1953,\\ \frac{63\cdot64}{2}&=2016 \end{aligned}\text{。}$ 因此这本书有 $62$ 页，多加的一项为 $1986-1953=33$ ，它确实是一个有效页码。

Consecutive triangular numbers around $1986$ are $\begin{aligned} \frac{62\cdot63}{2}&=1953,\\ \frac{63\cdot64}{2}&=2016. \end{aligned}$ Hence the book has $62$ pages, and the extra summand is $1986-1953=33,$ which is indeed a valid page number.

1986 AIME 第 6 题

答案：33

难度评级：1490

小提示：

大提示：

解答：

其他年份的第 6 题