1986 AIME Problems/Problem 6

Problem

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?

Solution

Denote the page number as $x$ , with $x < n$ . The sum formula shows that $\frac{n(n + 1)}{2} + x = 1986$ . Since $x$ cannot be very large, disregard it for now and solve $\frac{n(n+1)}{2} = 1986$ . The positive root for $n \approx \sqrt{3972} \approx 63$ . Quickly testing, we find that $63$ is too large, but if we plug in $62$ we find that $\frac{62(63)}{2} + x = 1986 \Longrightarrow x = 33$ , our solution.

1986 AIME (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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1986 AIME Problems/Problem 6

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