2015 AIME II Problems/Problem 13
Problem
Define the sequence by
, where
represents radian measure. Find the index of the 100th term for which
.
Solution
If ,
. Then if
satisfies
,
, and
Since
is positive, it does not affect the sign of
. Let
. Now since
and
,
is negative if and only if
, or when
. Since
is irrational, there is always only one integer in the range, so there are values of
such that
at
. Then the hundredth such value will be when
and
.
See also
2015 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.