1989 AIME Problems/Problem 1

Problem

Compute $\sqrt{(31)(30)(29)(28)+1}$ .

Solution

Let's call our four consecutive integers $(n-1), n, (n+1), (n+2)$ . Notice that $(n-1)(n)(n+1)(n+2)+1=(n^2+n)^2-2(n^2+n)+1 \Rightarrow (n^2+n-1)^2$ . Thus, $\sqrt{(31)(30)(29)(28)+1} = (29^2+29-1) = \boxed{869}$ .

1989 AIME (Problems • Answer Key • Resources)
Preceded by First Question	Followed by Problem 2
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1989 AIME Problems/Problem 1

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