2014 AMC 12A Problems/Problem 24
Problem
Let , and for
, let
. For how many values of
is
?
Solution 1
1. Draw the graph of by dividing the domain into three parts.
2. Apply the recursive rule a few times to find the pattern.
Note: is used to enlarge the difference, but the reasoning is the same.
3. Extrapolate to . Notice that the summits start
away from
and get
closer each iteration, so they reach
exactly at
.
reaches
at
, then zigzags between
and
, hitting
at every even
, before leaving
at
.
This means that at all even
where
. This is a
-integer odd-size range with even numbers at the endpoints, so just over half of the integers are even, or
.
(Revised by Flamedragon & Jason,C & emerald_block)
Solution 2
First, notice that the recursion and the definition of require that for all
such that
, if
, then
is even. Now, we can do case work on
to find which values of
(such that
) make
even. The answer comes out to be all the even values of
in the range
, in the domain
. So, the answer is
or
.
Video Solution by Richard Rusczyk
https://artofproblemsolving.com/videos/amc/2014amc12a/383
~ dolphin7
See Also
2014 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 23 |
Followed by Problem 25 |
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