2000 AIME I Problems/Problem 12
From AoPSWiki
Problem
Given a function 
for which
holds for all real 
what is the largest number of different values that can appear in the list 
Solution
Since 
we can conclude that (by the Euclidean algorithm)
So we need only to consider one period 
, which can have at most 
distinct values which determine the value of 
at all other integers.
But we also know that 
, so the values 
and 
are repeated. This gives a total of
distinct values.
To show that it is possible to have 
distinct, we try to find a function which fulfills the given conditions. A bit of trial and error would lead to the cosine function: 
(in degrees).
See also
| 2000 AIME I (Problems • Resources) | ||
| Preceded by Problem 11 | Followed by Problem 13 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
