Difference between revisions of "Mock AIME 4 2006-2007 Problems/Problem 14"
| (One intermediate revision by one other user not shown) | |||
| Line 3: | Line 3: | ||
==Solution== | ==Solution== | ||
| − | + | We will assume that there is at least one solution, otherwise the answer would be undefined. | |
| + | Using the binomial theorem it is obvious that <math>(577-k)^4 \equiv k^4 \pmod {577}</math>. Thus the solutions come in pairs <math>\{k,577-k\}</math>, and hence their average is <math>\dfrac{577}2 = 288.5</math>, and the answer is <math>\boxed{288}</math>. | ||
| + | |||
| + | (In this case, there are four solutions: <math>276</math>, <math>277</math>, <math>300</math>, and <math>301</math>.) | ||
| + | |||
| + | ---- | ||
| + | |||
| + | *[[Mock AIME 4 2006-2007 Problems/Problem 15| Next Problem]] | ||
| + | *[[Mock AIME 4 2006-2007 Problems/Problem 13| Previous Problem]] | ||
*[[Mock AIME 4 2006-2007 Problems]] | *[[Mock AIME 4 2006-2007 Problems]] | ||
Latest revision as of 01:52, 31 January 2009
Problem
Let 
be the arithmetic mean of all positive integers 
such that
.
Find the greatest integer less than or equal to .
Solution
We will assume that there is at least one solution, otherwise the answer would be undefined.
Using the binomial theorem it is obvious that 
. Thus the solutions come in pairs 
, and hence their average is 
, and the answer is 
.
(In this case, there are four solutions: 
, 
, 
, and 
.)
