1984 AIME Problems/Problem 12
A function is defined for all real numbers and satisfies and for all . If is a root for , what is the least number of roots must have in the interval ?
If , then substituting gives . Similarly, . In particular,
Since 0 is a root, all multiples of 10 are roots, and anything of the form "4 minus a multiple of 10" (that is, anything congruent to 4 modulo 10) are also roots. To see that these may be the only integer roots, observe that the function
satisfies the conditions and has no other roots.
In the interval , there are 201 multiples of 10 and 200 numbers that are congruent to 4 modulo 10, therefore the minimum number of roots is