Difference between revisions of "1994 AIME Problems/Problem 3"
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== Problem == | == Problem == | ||
The function <math>f_{}^{}</math> has the property that, for each real number <math>x,\,</math> | The function <math>f_{}^{}</math> has the property that, for each real number <math>x,\,</math> | ||
| − | <center><math>f(x)+f(x-1) = x^2\,</math></center> | + | <center><math>f(x)+f(x-1) = x^2.\,</math></center> |
If <math>f(19)=94,\,</math> what is the remainder when <math>f(94)\,</math> is divided by <math>1000</math>? | If <math>f(19)=94,\,</math> what is the remainder when <math>f(94)\,</math> is divided by <math>1000</math>? | ||
Revision as of 13:58, 13 June 2021
Contents
Problem
The function 
has the property that, for each real number 
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,
![\[T_{n-1} + T_n = n^2,\]](http://latex.artofproblemsolving.com/f/9/d/f9de2976d81b29dcd19878168f2cf4a6d14a5600.png)
where 
is the 
th triangular number.
Using this, as well as using the fact that the value of 
directly determines the value of 
and 
we conclude that 
for all odd and 
for all even 
where 
is a constant real number.
Since 
and 
we see that 
It follows that 
so the answer is 
See also
| 1994 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
