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

1989 AIME (Problems • Answer Key • Resources)
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

Page

Toolbox

Search

1989 AIME Problems/Problem 1

Problem

Solution

See also