1986 AIME Problem 6

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

Compare

1986

with consecutive triangular numbers

The excess over

1+2+\cdots+n

is the repeated page number

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.

Problem 6 in other years