1984 AIME Problems/Problem 12
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Problem
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 
?
Solution
If , then substituting 
gives 
. Similarly, 
. In particular,
Since 
is a root, all multiples of 
are roots, and anything congruent to 
) 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 
multiples of and 
numbers that are congruent to , therefore the minimum number of roots is 
.
See also
| 1984 AIME (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 | ||
