Difference between revisions of "1998 AJHSME Problems/Problem 18"
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| − | ==Problem | + | ==Problem== |
As indicated by the diagram below, a rectangular piece of paper is folded bottom to top, then left to right, and finally, a hole is punched at X. What does the paper look like when unfolded? | As indicated by the diagram below, a rectangular piece of paper is folded bottom to top, then left to right, and finally, a hole is punched at X. What does the paper look like when unfolded? | ||
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</asy> | </asy> | ||
| − | ==Solution 1== | + | ==Solution== |
| + | ===Solution 1=== | ||
By reversing the folds, we find that the holes will be | By reversing the folds, we find that the holes will be | ||
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This is pictured in <math>\boxed{B}</math> | This is pictured in <math>\boxed{B}</math> | ||
| − | ==Solution 2== | + | ===Solution 2=== |
If you unfold the paper, the hole you punched will not move. The only answer that keeps the hole in the upper-right quadrant in the same place is <math>\boxed{B}</math>. | If you unfold the paper, the hole you punched will not move. The only answer that keeps the hole in the upper-right quadrant in the same place is <math>\boxed{B}</math>. | ||
Revision as of 12:57, 23 December 2012
Problem
As indicated by the diagram below, a rectangular piece of paper is folded bottom to top, then left to right, and finally, a hole is punched at X. What does the paper look like when unfolded?
![[asy] draw((2,0)--(2,1)--(4,1)--(4,0)--cycle); draw(circle((2.25,.75),.225)); draw((2.05,.95)--(2.45,.55)); draw((2.45,.95)--(2.05,.55)); draw((0,2)--(4,2)--(4,3)--(0,3)--cycle); draw((2,2)--(2,3),dashed); draw((1.3,2.1)..(2,2.3)..(2.7,2.1),EndArrow); draw((1.3,3.1)..(2,3.3)..(2.7,3.1),EndArrow); draw((0,4)--(4,4)--(4,6)--(0,6)--cycle); draw((0,5)--(4,5),dashed); draw((-.1,4.3)..(-.3,5)..(-.1,5.7),EndArrow); draw((3.9,4.3)..(3.7,5)..(3.9,5.7),EndArrow); [/asy]](http://latex.artofproblemsolving.com/c/0/4/c04ada3dd179826a4afba783466e517d385634c6.png)
![[asy] unitsize(5); draw((0,0)--(16,0)--(16,8)--(0,8)--cycle); draw((0,4)--(16,4),dashed); draw((8,0)--(8,8),dashed); draw(circle((1,3),.9)); draw(circle((7,7),.9)); draw(circle((15,5),.9)); draw(circle((9,1),.9)); draw((24,0)--(40,0)--(40,8)--(24,8)--cycle); draw((24,4)--(40,4),dashed); draw((32,0)--(32,8),dashed); draw(circle((31,1),.9)); draw(circle((33,1),.9)); draw(circle((31,7),.9)); draw(circle((33,7),.9)); draw((48,0)--(64,0)--(64,8)--(48,8)--cycle); draw((48,4)--(64,4),dashed); draw((56,0)--(56,8),dashed); draw(circle((49,1),.9)); draw(circle((49,7),.9)); draw(circle((63,1),.9)); draw(circle((63,7),.9)); draw((72,0)--(88,0)--(88,8)--(72,8)--cycle); draw((72,4)--(88,4),dashed); draw((80,0)--(80,8),dashed); draw(circle((79,3),.9)); draw(circle((79,5),.9)); draw(circle((81,3),.9)); draw(circle((81,5),.9)); draw((96,0)--(112,0)--(112,8)--(96,8)--cycle); draw((96,4)--(112,4),dashed); draw((104,0)--(104,8),dashed); draw(circle((97,3),.9)); draw(circle((97,5),.9)); draw(circle((111,3),.9)); draw(circle((111,5),.9)); label("(A)",(8,10),N); label("(B)",(32,10),N); label("(C)",(56,10),N); label("(D)",(80,10),N); label("(E)",(104,10),N); [/asy]](http://latex.artofproblemsolving.com/f/2/d/f2ded21972617c562dcb820e97ef46969a85af3a.png)
Solution
Solution 1
By reversing the folds, we find that the holes will be
1) In the middle (in relation from left to right)
2) At separating poles (up and down)
This is pictured in 
Solution 2
If you unfold the paper, the hole you punched will not move. The only answer that keeps the hole in the upper-right quadrant in the same place is .
See also
| 1998 AJHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 17 |
Followed by Problem 19 | |
| 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 | ||
