Difference between revisions of "2023 AMC 8 Problems/Problem 2"
Themathguyd (talk | contribs) (→Problem: I have placed asy code for the answers. Could someone please crop the main image until I am done with the paper folding&cutting diagram?) |
MRENTHUSIASM (talk | contribs) (→*Easy Video Explanation by MathTalks_Now*) |
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==Problem== | ==Problem== | ||
| − | + | A square piece of paper is folded twice into four equal quarters, as shown below, then cut along the dashed line. When unfolded, the paper will match which of the following figures? | |
| − | + | <asy> | |
| − | + | //Restored original diagram. Alter it if you would like, but it was made by TheMathGuyd, | |
| − | + | // Diagram by TheMathGuyd. I even put the lined texture :) | |
| − | + | // Thank you Kante314 for inspiring thicker arrows. They do look much better | |
| + | size(0,3cm); | ||
| + | path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle; | ||
| + | path rh = (-0.125,-0.125)--(0.5,-0.5)--(0.5,0.5)--(-0.125,0.875)--cycle; | ||
path sqA = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,-0.25)--(0.25,0)--(0.5,0.25)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--(-0.5,0.25)--(-0.25,0)--(-0.5,-0.25)--cycle; | path sqA = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,-0.25)--(0.25,0)--(0.5,0.25)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--(-0.5,0.25)--(-0.25,0)--(-0.5,-0.25)--cycle; | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
path sqB = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--cycle; | path sqB = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--cycle; | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
path sqC = (-0.25,-0.25)--(0.25,-0.25)--(0.25,0.25)--(-0.25,0.25)--cycle; | path sqC = (-0.25,-0.25)--(0.25,-0.25)--(0.25,0.25)--(-0.25,0.25)--cycle; | ||
| + | path trD = (-0.25,0)--(0.25,0)--(0,0.25)--cycle; | ||
| + | path sqE = (-0.25,0)--(0,-0.25)--(0.25,0)--(0,0.25)--cycle; | ||
filldraw(sq,mediumgrey,black); | filldraw(sq,mediumgrey,black); | ||
| − | filldraw(sqC,white,black); | + | draw((0.75,0)--(1.25,0),currentpen+1,Arrow(size=6)); |
| + | //folding | ||
| + | path sqside = (-0.5,-0.5)--(0.5,-0.5); | ||
| + | path rhside = (-0.125,-0.125)--(0.5,-0.5); | ||
| + | transform fld = shift((1.75,0))*scale(0.5); | ||
| + | draw(fld*sq,black); | ||
| + | int i; | ||
| + | for(i=0; i<10; i=i+1) | ||
| + | { | ||
| + | draw(shift(0,0.05*i)*fld*sqside,deepblue); | ||
| + | } | ||
| + | path rhedge = (-0.125,-0.125)--(-0.125,0.8)--(-0.2,0.85)--cycle; | ||
| + | filldraw(fld*rhedge,grey); | ||
| + | path sqedge = (-0.5,-0.5)--(-0.5,0.4475)--(-0.575,0.45)--cycle; | ||
| + | filldraw(fld*sqedge,grey); | ||
| + | filldraw(fld*rh,white,black); | ||
| + | int i; | ||
| + | for(i=0; i<10; i=i+1) | ||
| + | { | ||
| + | draw(shift(0,0.05*i)*fld*rhside,deepblue); | ||
| + | } | ||
| + | draw((2.25,0)--(2.75,0),currentpen+1,Arrow(size=6)); | ||
| + | //cutting | ||
| + | transform cut = shift((3.25,0))*scale(0.5); | ||
| + | draw(shift((-0.01,+0.01))*cut*sq); | ||
| + | draw(cut*sq); | ||
| + | filldraw(shift((0.01,-0.01))*cut*sq,white,black); | ||
| + | int j; | ||
| + | for(j=0; j<10; j=j+1) | ||
| + | { | ||
| + | draw(shift(0,0.05*j)*cut*sqside,deepblue); | ||
| + | } | ||
| + | draw(shift((0.01,-0.01))*cut*(0,-0.5)--shift((0.01,-0.01))*cut*(0.5,0),dashed); | ||
| + | //Answers Below, but already Separated | ||
| + | //filldraw(sqA,grey,black); | ||
| + | //filldraw(sqB,grey,black); | ||
| + | //filldraw(sq,grey,black); | ||
| + | //filldraw(sqC,white,black); | ||
| + | //filldraw(sq,grey,black); | ||
| + | //filldraw(trD,white,black); | ||
| + | //filldraw(sq,grey,black); | ||
| + | //filldraw(sqE,white,black); | ||
</asy> | </asy> | ||
| − | < | + | <asy> |
| − | size( | + | // Diagram by TheMathGuyd. |
| + | size(0,7.5cm); | ||
path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle; | path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle; | ||
| + | path rh = (-0.125,-0.125)--(0.5,-0.5)--(0.5,0.5)--(-0.125,0.875)--cycle; | ||
| + | path sqA = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,-0.25)--(0.25,0)--(0.5,0.25)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--(-0.5,0.25)--(-0.25,0)--(-0.5,-0.25)--cycle; | ||
| + | path sqB = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--cycle; | ||
| + | path sqC = (-0.25,-0.25)--(0.25,-0.25)--(0.25,0.25)--(-0.25,0.25)--cycle; | ||
path trD = (-0.25,0)--(0.25,0)--(0,0.25)--cycle; | path trD = (-0.25,0)--(0.25,0)--(0,0.25)--cycle; | ||
| − | filldraw(sq,mediumgrey,black); | + | path sqE = (-0.25,0)--(0,-0.25)--(0.25,0)--(0,0.25)--cycle; |
| − | filldraw(trD,white,black); | + | |
| + | //ANSWERS | ||
| + | real sh = 1.5; | ||
| + | label("$\textbf{(A)}$",(-0.5,0.5),SW); | ||
| + | label("$\textbf{(B)}$",shift((sh,0))*(-0.5,0.5),SW); | ||
| + | label("$\textbf{(C)}$",shift((2sh,0))*(-0.5,0.5),SW); | ||
| + | label("$\textbf{(D)}$",shift((0,-sh))*(-0.5,0.5),SW); | ||
| + | label("$\textbf{(E)}$",shift((sh,-sh))*(-0.5,0.5),SW); | ||
| + | filldraw(sqA,mediumgrey,black); | ||
| + | filldraw(shift((sh,0))*sqB,mediumgrey,black); | ||
| + | filldraw(shift((2*sh,0))*sq,mediumgrey,black); | ||
| + | filldraw(shift((2*sh,0))*sqC,white,black); | ||
| + | filldraw(shift((0,-sh))*sq,mediumgrey,black); | ||
| + | filldraw(shift((0,-sh))*trD,white,black); | ||
| + | filldraw(shift((sh,-sh))*sq,mediumgrey,black); | ||
| + | filldraw(shift((sh,-sh))*sqE,white,black); | ||
</asy> | </asy> | ||
| − | < | + | ==Solution== |
| − | size( | + | |
| + | Notice that when we unfold the paper along the vertical fold line, we get the following shape: | ||
| + | |||
| + | |||
| + | <asy> | ||
| + | |||
| + | size(90); | ||
| + | path sq = (-0.5,0)--(0.5,0)--(0.5,0.5)--(-0.5,0.5)--cycle; | ||
| + | path trE = (-0.25,0)--(0.25,0)--(0,0.25)--cycle; | ||
| + | |||
| + | real sh = 1.5; | ||
| + | filldraw(shift((sh,-sh))*sq,mediumgrey,black); | ||
| + | filldraw(shift((sh,-sh))*trE,white,black); | ||
| + | |||
| + | size(90); | ||
path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle; | path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle; | ||
path sqE = (-0.25,0)--(0,-0.25)--(0.25,0)--(0,0.25)--cycle; | path sqE = (-0.25,0)--(0,-0.25)--(0.25,0)--(0,0.25)--cycle; | ||
| − | filldraw(sq,mediumgrey,black); | + | |
| − | filldraw(sqE,white,black); | + | real sh = 1.5; |
| + | filldraw(shift((sh,-sh))*sq,mediumgrey,black); | ||
| + | filldraw(shift((sh,-sh))*sqE,white,black); | ||
</asy> | </asy> | ||
| − | |||
| − | + | It is clear that the answer is <math>\boxed{\textbf{(E)}}</math>. | |
| − | + | ~MrThinker | |
| − | ==Solution 2 | + | ==Solution 2== |
| − | + | When you fold the paper in that specific way, it shows the top left square. That means that any cut you make on that folded part will look like that on that top left square. Since it is folded into four parts, the cut will reflect on all of the other 3 parts. Since there is a cut diagonally on the bottom right of that square, that will reflect on the other squares, making the shape of <math>\boxed{\textbf{(E)}}</math> <asy> | |
| − | + | size(90); | |
| + | path sq = (-0.5,0)--(0.5,0)--(0.5,0.5)--(-0.5,0.5)--cycle; | ||
| + | path trE = (-0.25,0)--(0.25,0)--(0,0.25)--cycle; | ||
| − | + | real sh = 1.5; | |
| + | filldraw(shift((sh,-sh))*sq,mediumgrey,black); | ||
| + | filldraw(shift((sh,-sh))*trE,white,black); | ||
| − | + | size(90); | |
| + | path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle; | ||
| + | path sqE = (-0.25,0)--(0,-0.25)--(0.25,0)--(0,0.25)--cycle; | ||
| − | + | real sh = 1.5; | |
| + | filldraw(shift((sh,-sh))*sq,mediumgrey,black); | ||
| + | filldraw(shift((sh,-sh))*sqE,white,black); | ||
| + | </asy> | ||
| − | ~ | + | ~AliceDubbleYou |
| + | |||
| + | ==*Easy Video Explanation by MathTalks_Now*== | ||
| + | https://studio.youtube.com/video/PMOeiGLkDH0/edit | ||
| + | |||
| + | ==Video Solution by Math-X (Smart and Simple)== | ||
| + | https://youtu.be/Ku_c1YHnLt0?si=ZucTBcN42MKGX2Ty&t=115 ~Math-X | ||
| + | |||
| + | ==Video Solution (How to Creatively THINK!!!)== | ||
| + | https://youtu.be/suFxwnH-ak8 | ||
| + | ~Education the Study of everything | ||
==Video Solution by Magic Square== | ==Video Solution by Magic Square== | ||
https://youtu.be/-N46BeEKaCQ?t=5658 | https://youtu.be/-N46BeEKaCQ?t=5658 | ||
| + | |||
| + | ==Video Solution by SpreadTheMathLove== | ||
| + | https://www.youtube.com/watch?v=EcrktBc8zrM | ||
| + | ==Video Solution by Interstigation== | ||
| + | https://youtu.be/DBqko2xATxs&t=67 | ||
| + | |||
| + | ==Video Solution by WhyMath== | ||
| + | |||
| + | https://youtu.be/z6SxQkQACjo?si=WJAMIdKzUO7oGLGc | ||
| + | |||
| + | ==Video Solution by harungurcan== | ||
| + | https://www.youtube.com/watch?v=35BW7bsm_Cg&t=97s | ||
| + | |||
| + | ~harungurcan | ||
| + | |||
| + | ==Video Solution by Dr. David== | ||
| + | https://youtu.be/octW02FH-iU | ||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2023|num-b=1|num-a=3}} | {{AMC8 box|year=2023|num-b=1|num-a=3}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 01:53, 4 November 2024
Contents
- 1 Problem
- 2 Solution
- 3 Solution 2
- 4 *Easy Video Explanation by MathTalks_Now*
- 5 Video Solution by Math-X (Smart and Simple)
- 6 Video Solution (How to Creatively THINK!!!)
- 7 Video Solution by Magic Square
- 8 Video Solution by SpreadTheMathLove
- 9 Video Solution by Interstigation
- 10 Video Solution by WhyMath
- 11 Video Solution by harungurcan
- 12 Video Solution by Dr. David
- 13 See Also
Problem
A square piece of paper is folded twice into four equal quarters, as shown below, then cut along the dashed line. When unfolded, the paper will match which of the following figures?
![[asy] //Restored original diagram. Alter it if you would like, but it was made by TheMathGuyd, // Diagram by TheMathGuyd. I even put the lined texture :) // Thank you Kante314 for inspiring thicker arrows. They do look much better size(0,3cm); path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle; path rh = (-0.125,-0.125)--(0.5,-0.5)--(0.5,0.5)--(-0.125,0.875)--cycle; path sqA = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,-0.25)--(0.25,0)--(0.5,0.25)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--(-0.5,0.25)--(-0.25,0)--(-0.5,-0.25)--cycle; path sqB = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--cycle; path sqC = (-0.25,-0.25)--(0.25,-0.25)--(0.25,0.25)--(-0.25,0.25)--cycle; path trD = (-0.25,0)--(0.25,0)--(0,0.25)--cycle; path sqE = (-0.25,0)--(0,-0.25)--(0.25,0)--(0,0.25)--cycle; filldraw(sq,mediumgrey,black); draw((0.75,0)--(1.25,0),currentpen+1,Arrow(size=6)); //folding path sqside = (-0.5,-0.5)--(0.5,-0.5); path rhside = (-0.125,-0.125)--(0.5,-0.5); transform fld = shift((1.75,0))*scale(0.5); draw(fld*sq,black); int i; for(i=0; i<10; i=i+1) { draw(shift(0,0.05*i)*fld*sqside,deepblue); } path rhedge = (-0.125,-0.125)--(-0.125,0.8)--(-0.2,0.85)--cycle; filldraw(fld*rhedge,grey); path sqedge = (-0.5,-0.5)--(-0.5,0.4475)--(-0.575,0.45)--cycle; filldraw(fld*sqedge,grey); filldraw(fld*rh,white,black); int i; for(i=0; i<10; i=i+1) { draw(shift(0,0.05*i)*fld*rhside,deepblue); } draw((2.25,0)--(2.75,0),currentpen+1,Arrow(size=6)); //cutting transform cut = shift((3.25,0))*scale(0.5); draw(shift((-0.01,+0.01))*cut*sq); draw(cut*sq); filldraw(shift((0.01,-0.01))*cut*sq,white,black); int j; for(j=0; j<10; j=j+1) { draw(shift(0,0.05*j)*cut*sqside,deepblue); } draw(shift((0.01,-0.01))*cut*(0,-0.5)--shift((0.01,-0.01))*cut*(0.5,0),dashed); //Answers Below, but already Separated //filldraw(sqA,grey,black); //filldraw(sqB,grey,black); //filldraw(sq,grey,black); //filldraw(sqC,white,black); //filldraw(sq,grey,black); //filldraw(trD,white,black); //filldraw(sq,grey,black); //filldraw(sqE,white,black); [/asy]](http://latex.artofproblemsolving.com/6/d/3/6d385621437744e469b64fd20d5c8183e9e93ac1.png)
![[asy] // Diagram by TheMathGuyd. size(0,7.5cm); path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle; path rh = (-0.125,-0.125)--(0.5,-0.5)--(0.5,0.5)--(-0.125,0.875)--cycle; path sqA = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,-0.25)--(0.25,0)--(0.5,0.25)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--(-0.5,0.25)--(-0.25,0)--(-0.5,-0.25)--cycle; path sqB = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--cycle; path sqC = (-0.25,-0.25)--(0.25,-0.25)--(0.25,0.25)--(-0.25,0.25)--cycle; path trD = (-0.25,0)--(0.25,0)--(0,0.25)--cycle; path sqE = (-0.25,0)--(0,-0.25)--(0.25,0)--(0,0.25)--cycle; //ANSWERS real sh = 1.5; label("$\textbf{(A)}$",(-0.5,0.5),SW); label("$\textbf{(B)}$",shift((sh,0))*(-0.5,0.5),SW); label("$\textbf{(C)}$",shift((2sh,0))*(-0.5,0.5),SW); label("$\textbf{(D)}$",shift((0,-sh))*(-0.5,0.5),SW); label("$\textbf{(E)}$",shift((sh,-sh))*(-0.5,0.5),SW); filldraw(sqA,mediumgrey,black); filldraw(shift((sh,0))*sqB,mediumgrey,black); filldraw(shift((2*sh,0))*sq,mediumgrey,black); filldraw(shift((2*sh,0))*sqC,white,black); filldraw(shift((0,-sh))*sq,mediumgrey,black); filldraw(shift((0,-sh))*trD,white,black); filldraw(shift((sh,-sh))*sq,mediumgrey,black); filldraw(shift((sh,-sh))*sqE,white,black); [/asy]](http://latex.artofproblemsolving.com/0/f/8/0f8ca5448d7ae5672368a192ed9ee5af7a40419b.png)
Solution
Notice that when we unfold the paper along the vertical fold line, we get the following shape:
![[asy] size(90); path sq = (-0.5,0)--(0.5,0)--(0.5,0.5)--(-0.5,0.5)--cycle; path trE = (-0.25,0)--(0.25,0)--(0,0.25)--cycle; real sh = 1.5; filldraw(shift((sh,-sh))*sq,mediumgrey,black); filldraw(shift((sh,-sh))*trE,white,black); size(90); path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle; path sqE = (-0.25,0)--(0,-0.25)--(0.25,0)--(0,0.25)--cycle; real sh = 1.5; filldraw(shift((sh,-sh))*sq,mediumgrey,black); filldraw(shift((sh,-sh))*sqE,white,black); [/asy]](http://latex.artofproblemsolving.com/e/a/1/ea168c59de1838ccc582149a810ed68af3cb9940.png)
It is clear that the answer is 
.
~MrThinker
Solution 2
When you fold the paper in that specific way, it shows the top left square. That means that any cut you make on that folded part will look like that on that top left square. Since it is folded into four parts, the cut will reflect on all of the other 3 parts. Since there is a cut diagonally on the bottom right of that square, that will reflect on the other squares, making the shape of
~AliceDubbleYou
*Easy Video Explanation by MathTalks_Now*
https://studio.youtube.com/video/PMOeiGLkDH0/edit
Video Solution by Math-X (Smart and Simple)
https://youtu.be/Ku_c1YHnLt0?si=ZucTBcN42MKGX2Ty&t=115 ~Math-X
Video Solution (How to Creatively THINK!!!)
https://youtu.be/suFxwnH-ak8 ~Education the Study of everything
Video Solution by Magic Square
https://youtu.be/-N46BeEKaCQ?t=5658
Video Solution by SpreadTheMathLove
https://www.youtube.com/watch?v=EcrktBc8zrM
Video Solution by Interstigation
https://youtu.be/DBqko2xATxs&t=67
Video Solution by WhyMath
https://youtu.be/z6SxQkQACjo?si=WJAMIdKzUO7oGLGc
Video Solution by harungurcan
https://www.youtube.com/watch?v=35BW7bsm_Cg&t=97s
~harungurcan
Video Solution by Dr. David
See Also
| 2023 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
| 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 AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
