Difference between revisions of "2023 AMC 8 Problems/Problem 2"
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− | + | ==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> | ||
− | + | <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> | ||
+ | |||
+ | ==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> | ||
+ | |||
+ | It is clear that the answer is <math>\boxed{\textbf{(E)}}</math>. | ||
+ | ~MrThinker | ||
+ | |||
+ | ==*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== | ||
+ | |||
+ | ~savannahsolver | ||
+ | |||
+ | ==Video Solution by harungurcan== | ||
+ | https://www.youtube.com/watch?v=35BW7bsm_Cg&t=97s | ||
+ | |||
+ | ~harungurcan | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC8 box|year=2023|num-b=1|num-a=3}} | ||
+ | {{MAA Notice}} |
Latest revision as of 00:30, 19 May 2024
Contents
- 1 Problem
- 2 Solution
- 3 *Easy Video Explanation by MathTalks_Now*
- 4 Video Solution by Math-X (Smart and Simple)
- 5 Video Solution (How to Creatively THINK!!!)
- 6 Video Solution by Magic Square
- 7 Video Solution by SpreadTheMathLove
- 8 Video Solution by Interstigation
- 9 Video Solution by WhyMath
- 10 Video Solution by harungurcan
- 11 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?
Solution
Notice that when we unfold the paper along the vertical fold line, we get the following shape:
It is clear that the answer is .
~MrThinker
*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
~savannahsolver
Video Solution by harungurcan
https://www.youtube.com/watch?v=35BW7bsm_Cg&t=97s
~harungurcan
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.