1994 AIME Problems/Problem 3
Contents
Problem
The function has the property that, for each real number
![$f(x)+f(x-1) = x^2\,$](http://latex.artofproblemsolving.com/6/7/8/67827ba82e465559d945580a479bca7bcce2fdd9.png)
.
If what is the remainder when
is divided by
?
Solution 1
So, the remainder is .
Solution 2
Those familiar with triangular numbers and some of their properties will quickly recognize the equation given in the problem. It is well-known (and easy to show) that the sum of two consecutive triangular numbers is a perfect square; that is,
where
is the
th triangular number.
See also
1994 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
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