Difference between revisions of "2007 AIME I Problems/Problem 14"
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Revision as of 20:11, 4 July 2013
Problem
A sequence is defined over non-negative integral indexes in the following way: ,
.
Find the greatest integer that does not exceed
Solution
We are given that
,
.
Add these two equations to get
.
This is an invariant. Defining for each
, the above equation means
.
We can thus calculate that . Now notice that
. This means that
. It is only a tiny bit less because all the
are greater than
, so we conclude that the floor of
is
.
See also
2007 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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.