2000 AIME I Problems/Problem 12

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).

Solution 2

One can imagine that there must be multiple lines of symmetry for the function , as if a function can be expressed with it must be symmetric against line . Try this yourself by graphing a polynomial , then graphing . If , their point of intersection at must contain a line of symmetry.

   For this particular function , it has 3 other values

See also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.