Difference between revisions of "2014 AMC 12A Problems/Problem 24"
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3. Count the x intercepts of the each function and find the pattern. | 3. Count the x intercepts of the each function and find the pattern. | ||
− | The pattern turns out to be <math>3n+1</math> | + | The pattern turns out to be <math>3n+3</math> solutions,x interval:[1,99], the function gain only one extra solution after <math>f_99(x)</math> because there is no summits on the graph any more, and the answer is thus <math>\textbf{(C) }301\qquad</math>. |
− | (Revised by Flamedragon) | + | (Revised by Flamedragon & Jason,C) |
==Solution 2== | ==Solution 2== |
Revision as of 11:09, 30 July 2014
Contents
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. Look at the recursive rule. Take absolute of the previous function and down by 1 to get the next function.
3. Count the x intercepts of the each function and find the pattern.
The pattern turns out to be solutions,x interval:[1,99], the function gain only one extra solution after
because there is no summits on the graph any more, and the answer is thus
.
(Revised by Flamedragon & Jason,C)
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
. So, the answer is
or
.
See Also
2014 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 23 |
Followed by Problem 25 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.