Question

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?

Answer

Answer: 33

Concepts:bounding to limit cases triangular number

Difficulty rating: 1490

Small Hint:

Compare $1986$ with consecutive triangular numbers

Big Hint:

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

Solution:

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 Problem 6

Answer: 33

Difficulty rating: 1490

Small Hint:

Big Hint:

Solution:

Problem 6 in Other Years