1989 AIME Problems/Problem 1

Revision as of 13:56, 25 November 2007 by Azjps (talk | contribs) (c)

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, $\sqrt{(31)(30)(29)(28)+1} = (29^2+29-1) = \boxed{869}$.

See also

1989 AIME (ProblemsAnswer KeyResources)
Preceded by
First Question
Followed by
Problem 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions