Difference between revisions of "2023 AMC 8 Problems/Problem 2"

Revision as of 21:36, 16 February 2023

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: $[asy] /* Diagram by TheMathGuyd. Edited by MRENTHUSIASM */ size(90); path sq = (-0.5,0)--(0.5,0)--(0.5,0.5)--(-0.5,0.5)--cycle; path trE = (-0.25,0)--(0.25,0)--(0,0.25)--cycle; real sh = 1.5; filldraw(shift((sh,-sh))*sq,mediumgrey,black); filldraw(shift((sh,-sh))*trE,white,black); [/asy]$ Then, if we unfold the paper along the horizontal fold line, we get the following shape: $[asy] /* Diagram by TheMathGuyd. Edited by MRENTHUSIASM */ size(90); path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle; path sqE = (-0.25,0)--(0,-0.25)--(0.25,0)--(0,0.25)--cycle; real sh = 1.5; filldraw(shift((sh,-sh))*sq,mediumgrey,black); filldraw(shift((sh,-sh))*sqE,white,black); [/asy]$ It is clear that the answer is $\boxed{\textbf{(E)}}$ .

~MrThinker

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/1bA7fD7Lg54?t=52

@@ Line 126: / Line 126: @@
 https://www.youtube.com/watch?v=EcrktBc8zrM
 ==Video Solution by Interstigation==
-https://www.youtube.com/watch?v=1bA7fD7Lg54
+https://youtu.be/1bA7fD7Lg54?t=52
 ==See Also==
 {{AMC8 box|year=2023|num-b=1|num-a=3}}
 {{MAA Notice}}

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

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