Difference between revisions of "2021 Fall AMC 10A Problems/Problem 16"
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Revision as of 23:19, 29 November 2021
Contents
Problem
The graph of is symmetric about which of the following? (Here
is the greatest integer not exceeding
.)
Solution 1 (Graphing)
Let and
Note that the graph of
is a reflection of the graph of
about the
-axis, followed by a translation of
unit right.
The graph of is shown below:
The graph of
is shown below:
The graph of
is shown below:
Therefore, the graph of is symmetric about
~MRENTHUSIASM
Solution 2 (Casework)
For all and
note that:
and
We rewrite as
We apply casework to the value of
and
and
and
It follows that
It follows that
It follows that
It follows that
It follows that
It follows that
Together, we have
so the graph of
is symmetric about
Alternatively, we can eliminate and
once we finish with Case 3. This leaves us with
~MRENTHUSIASM
Solution 3 (Casework)
Denote , where
and
.
Hence,
is the integer part of
and
is the decimal part of
.
:
.
We have
:
.
We have
Therefore, the graph of is symmetric through the point
.
Therefore, the answer is .
~Steven Chen (www.professorchenedu.com)
See Also
2021 Fall AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.