Difference between revisions of "2021 Fall AMC 10A Problems/Problem 16"
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\textbf{(D) }\text{ the point }\left(\dfrac12, 0\right)\qquad \textbf{(E) }\text{the point }(1,0)</math> | \textbf{(D) }\text{ the point }\left(\dfrac12, 0\right)\qquad \textbf{(E) }\text{the point }(1,0)</math> | ||
− | ==Solution 1 (Graphing)== | + | ==Solution 1== |
+ | Notice <math>f(1-x)=|\lfloor 1-x\rfloor|-|\lfloor x\rfloor|=-f(x)</math> so <math>f(1/2+x)=-f(1/2-x)</math>. This means that the graph is symmetric about <math>\boxed{\textbf{(D) }\text{ the point }\left(\frac12,0\right)}</math>. | ||
+ | |||
+ | ==Solution 2 (Graphing)== | ||
Let <math>y_1=|\lfloor x \rfloor|</math> and <math>y_2=|\lfloor 1 - x \rfloor|=|\lfloor -(x-1) \rfloor|.</math> Note that the graph of <math>y_2</math> is a reflection of the graph of <math>y_1</math> about the <math>y</math>-axis, followed by a translation <math>1</math> unit to the right. | Let <math>y_1=|\lfloor x \rfloor|</math> and <math>y_2=|\lfloor 1 - x \rfloor|=|\lfloor -(x-1) \rfloor|.</math> Note that the graph of <math>y_2</math> is a reflection of the graph of <math>y_1</math> about the <math>y</math>-axis, followed by a translation <math>1</math> unit to the right. | ||
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~MRENTHUSIASM | ~MRENTHUSIASM | ||
− | ==Solution | + | ==Solution 3 (Casework)== |
For all <math>x\in\mathbb{R}</math> and <math>n\in\mathbb{Z},</math> note that: | For all <math>x\in\mathbb{R}</math> and <math>n\in\mathbb{Z},</math> note that: | ||
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− | == Solution | + | == Solution 4 (Casework) == |
Denote <math>x = a + b</math>, where <math>a \in \Bbb Z</math> and <math>b \in \left[ 0 , 1 \right)</math>. | Denote <math>x = a + b</math>, where <math>a \in \Bbb Z</math> and <math>b \in \left[ 0 , 1 \right)</math>. | ||
Hence, <math>a</math> is the integer part of <math>x</math> and <math>b</math> is the decimal part of <math>x</math>. | Hence, <math>a</math> is the integer part of <math>x</math> and <math>b</math> is the decimal part of <math>x</math>. |
Revision as of 12:27, 22 October 2022
Contents
Problem
The graph of is symmetric about which of the following? (Here
is the greatest integer not exceeding
.)
Solution 1
Notice so
. This means that the graph is symmetric about
.
Solution 2 (Graphing)
Let and
Note that the graph of
is a reflection of the graph of
about the
-axis, followed by a translation
unit to the 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 3 (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 4 (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.