1984 AIME Problems/Problem 12
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 • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |