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2009 AMC 8 Problems/Problem 4: Difference between revisions

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==Problem==
==Problem==


The five pieces shown below can be arranged to form four of the five figures shown in the choices. Which figure '''cannot''' be formed?
The five pieces shown below can be arranged to form four of the five figures shown in the choices. Which figure <b>cannot</b> be formed?
 
<asy>
<asy>
defaultpen(linewidth(0.6));
defaultpen(linewidth(0.6));
Line 22: Line 21:
draw(shift(4s,0)*p);
draw(shift(4s,0)*p);
draw(shift(4s,2r)*p);
draw(shift(4s,2r)*p);
draw(shift(4s,4r)*p);</asy>
draw(shift(4s,4r)*p);
</asy>


<asy>
<asy>
Line 46: Line 46:


==Solution==
==Solution==
The answer is <math>\boxed{\textbf{(B)}}</math> because the longest piece can't fit in the figure.
The answer is <math>\boxed{\textbf{(B)}}</math> because the longest piece cannot fit into the figure.
 
Note that the five pieces can be arranged to form the figures in <math>\textbf{(A)},\textbf{(C)},\textbf{(D)},</math> and <math>\textbf{(E)},</math> as shown below:
 
<asy>
defaultpen(linewidth(0.6));
size(80);
real r=0.5, s=1.5;
path p=origin--(1,0)--(1,1)--(0,1)--cycle;
fill(p,red);
fill(shift(s,r)*p,yellow);
fill(shift(s,-r)*p,yellow);
fill(shift(2s,2r)*p,green);
fill(shift(2s,0)*p,green);
fill(shift(2s,-2r)*p,green);
fill(shift(3s,3r)*p,cyan);
fill(shift(3s,-3r)*p,cyan);
fill(shift(3s,r)*p,cyan);
fill(shift(3s,-r)*p,cyan);
fill(shift(4s,-4r)*p,pink);
fill(shift(4s,-2r)*p,pink);
fill(shift(4s,0)*p,pink);
fill(shift(4s,2r)*p,pink);
fill(shift(4s,4r)*p,pink);
draw(p);
draw(shift(s,r)*p);
draw(shift(s,-r)*p);
draw(shift(2s,2r)*p);
draw(shift(2s,0)*p);
draw(shift(2s,-2r)*p);
draw(shift(3s,3r)*p);
draw(shift(3s,-3r)*p);
draw(shift(3s,r)*p);
draw(shift(3s,-r)*p);
draw(shift(4s,-4r)*p);
draw(shift(4s,-2r)*p);
draw(shift(4s,0)*p);
draw(shift(4s,2r)*p);
draw(shift(4s,4r)*p);
</asy>
 
<asy>
size(350);
defaultpen(linewidth(0.6));
path p=origin--(1,0)--(1,1)--(0,1)--cycle;
pair[] a={(0,0), (0,1), (0,2), (0,3), (0,4), (1,0), (1,1), (1,2), (2,0), (2,1), (3,0), (3,1), (3,2), (3,3), (3,4)};
pair[] b={(5,3), (5,4), (6,2), (6,3), (6,4), (7,1), (7,2), (7,3), (7,4), (8,0), (8,1), (8,2), (9,0), (9,1), (9,2)};
pair[] c={(11,0), (11,1), (11,2), (11,3), (11,4), (12,1), (12,2), (12,3), (12,4), (13,2), (13,3), (13,4), (14,3), (14,4), (15,4)};
pair[] d={(17,0), (17,1), (17,2), (17,3), (17,4), (18,0), (18,1), (18,2), (18,3), (18,4), (19,0), (19,1), (19,2), (19,3), (19,4)};
pair[] e={(21,4), (22,1), (22,2), (22,3), (22,4), (23,0), (23,1), (23,2), (23,3), (23,4), (24,1), (24,2), (24,3), (24,4), (25,4)};
 
fill((0,0)--(0,5)--(1,5)--(1,0)--cycle,pink);
fill((1,0)--(1,3)--(2,3)--(2,0)--cycle,green);
fill((2,0)--(2,2)--(3,2)--(3,0)--cycle,yellow);
fill((3,4)--(3,5)--(4,5)--(4,4)--cycle,red);
fill((3,0)--(3,4)--(4,4)--(4,0)--cycle,cyan);
 
fill((11,0)--(11,5)--(12,5)--(12,0)--cycle,pink);
fill((12,1)--(12,5)--(13,5)--(13,1)--cycle,cyan);
fill((13,2)--(13,5)--(14,5)--(14,2)--cycle,green);
fill((14,3)--(14,5)--(15,5)--(15,3)--cycle,yellow);
fill((15,4)--(15,5)--(16,5)--(16,4)--cycle,red);
 
fill((17,0)--(17,5)--(18,5)--(18,0)--cycle,pink);
fill((18,0)--(18,4)--(19,4)--(19,0)--cycle,cyan);
fill((18,4)--(18,5)--(19,5)--(19,4)--cycle,red);
fill((19,0)--(19,3)--(20,3)--(20,0)--cycle,green);
fill((19,3)--(19,5)--(20,5)--(20,3)--cycle,yellow);
 
fill((21,4)--(21,5)--(22,5)--(22,4)--cycle,red);
fill((22,1)--(22,5)--(23,5)--(23,1)--cycle,cyan);
fill((23,0)--(23,5)--(24,5)--(24,0)--cycle,pink);
fill((24,1)--(24,4)--(25,4)--(25,1)--cycle,green);
fill((24,4)--(24,5)--(26,5)--(26,4)--cycle,yellow);
 
int i;
for(int i=0; i<15; i=i+1) {
draw(shift(a[i])*p);
draw(shift(b[i])*p);
draw(shift(c[i])*p);
draw(shift(d[i])*p);
draw(shift(e[i])*p);
}
</asy>
<cmath>\textbf{(A)}\qquad\qquad\qquad\textbf{(B)}\quad\qquad\qquad\textbf{(C)}\qquad\qquad\qquad\textbf{(D)}\quad\qquad\qquad\textbf{(E)}</cmath>
~Basketball8 ~MRENTHUSIASM
 
== Video Solution ==
https://youtu.be/USVVURBLaAc?t=171


==See Also==
==See Also==
{{AMC8 box|year=2009|num-b=3|num-a=5}}
{{AMC8 box|year=2009|num-b=3|num-a=5}}
{{MAA Notice}}
{{MAA Notice}}

Latest revision as of 21:44, 25 November 2024

Problem

The five pieces shown below can be arranged to form four of the five figures shown in the choices. Which figure cannot be formed? [asy] defaultpen(linewidth(0.6)); size(80); real r=0.5, s=1.5; path p=origin--(1,0)--(1,1)--(0,1)--cycle; draw(p); draw(shift(s,r)*p); draw(shift(s,-r)*p); draw(shift(2s,2r)*p); draw(shift(2s,0)*p); draw(shift(2s,-2r)*p); draw(shift(3s,3r)*p); draw(shift(3s,-3r)*p); draw(shift(3s,r)*p); draw(shift(3s,-r)*p); draw(shift(4s,-4r)*p); draw(shift(4s,-2r)*p); draw(shift(4s,0)*p); draw(shift(4s,2r)*p); draw(shift(4s,4r)*p); [/asy]

[asy] size(350); defaultpen(linewidth(0.6)); path p=origin--(1,0)--(1,1)--(0,1)--cycle; pair[] a={(0,0), (0,1), (0,2), (0,3), (0,4), (1,0), (1,1), (1,2), (2,0), (2,1), (3,0), (3,1), (3,2), (3,3), (3,4)}; pair[] b={(5,3), (5,4), (6,2), (6,3), (6,4), (7,1), (7,2), (7,3), (7,4), (8,0), (8,1), (8,2), (9,0), (9,1), (9,2)}; pair[] c={(11,0), (11,1), (11,2), (11,3), (11,4), (12,1), (12,2), (12,3), (12,4), (13,2), (13,3), (13,4), (14,3), (14,4), (15,4)}; pair[] d={(17,0), (17,1), (17,2), (17,3), (17,4), (18,0), (18,1), (18,2), (18,3), (18,4), (19,0), (19,1), (19,2), (19,3), (19,4)}; pair[] e={(21,4), (22,1), (22,2), (22,3), (22,4), (23,0), (23,1), (23,2), (23,3), (23,4), (24,1), (24,2), (24,3), (24,4), (25,4)}; int i; for(int i=0; i<15; i=i+1) { draw(shift(a[i])*p); draw(shift(b[i])*p); draw(shift(c[i])*p); draw(shift(d[i])*p); draw(shift(e[i])*p); } [/asy] \[\textbf{(A)}\qquad\qquad\qquad\textbf{(B)}\quad\qquad\qquad\textbf{(C)}\:\qquad\qquad\qquad\textbf{(D)}\quad\qquad\qquad\textbf{(E)}\]

Solution

The answer is $\boxed{\textbf{(B)}}$ because the longest piece cannot fit into the figure.

Note that the five pieces can be arranged to form the figures in $\textbf{(A)},\textbf{(C)},\textbf{(D)},$ and $\textbf{(E)},$ as shown below:

[asy] defaultpen(linewidth(0.6)); size(80); real r=0.5, s=1.5; path p=origin--(1,0)--(1,1)--(0,1)--cycle; fill(p,red); fill(shift(s,r)*p,yellow); fill(shift(s,-r)*p,yellow); fill(shift(2s,2r)*p,green); fill(shift(2s,0)*p,green); fill(shift(2s,-2r)*p,green); fill(shift(3s,3r)*p,cyan); fill(shift(3s,-3r)*p,cyan); fill(shift(3s,r)*p,cyan); fill(shift(3s,-r)*p,cyan); fill(shift(4s,-4r)*p,pink); fill(shift(4s,-2r)*p,pink); fill(shift(4s,0)*p,pink); fill(shift(4s,2r)*p,pink); fill(shift(4s,4r)*p,pink); draw(p); draw(shift(s,r)*p); draw(shift(s,-r)*p); draw(shift(2s,2r)*p); draw(shift(2s,0)*p); draw(shift(2s,-2r)*p); draw(shift(3s,3r)*p); draw(shift(3s,-3r)*p); draw(shift(3s,r)*p); draw(shift(3s,-r)*p); draw(shift(4s,-4r)*p); draw(shift(4s,-2r)*p); draw(shift(4s,0)*p); draw(shift(4s,2r)*p); draw(shift(4s,4r)*p); [/asy]

[asy] size(350); defaultpen(linewidth(0.6)); path p=origin--(1,0)--(1,1)--(0,1)--cycle; pair[] a={(0,0), (0,1), (0,2), (0,3), (0,4), (1,0), (1,1), (1,2), (2,0), (2,1), (3,0), (3,1), (3,2), (3,3), (3,4)}; pair[] b={(5,3), (5,4), (6,2), (6,3), (6,4), (7,1), (7,2), (7,3), (7,4), (8,0), (8,1), (8,2), (9,0), (9,1), (9,2)}; pair[] c={(11,0), (11,1), (11,2), (11,3), (11,4), (12,1), (12,2), (12,3), (12,4), (13,2), (13,3), (13,4), (14,3), (14,4), (15,4)}; pair[] d={(17,0), (17,1), (17,2), (17,3), (17,4), (18,0), (18,1), (18,2), (18,3), (18,4), (19,0), (19,1), (19,2), (19,3), (19,4)}; pair[] e={(21,4), (22,1), (22,2), (22,3), (22,4), (23,0), (23,1), (23,2), (23,3), (23,4), (24,1), (24,2), (24,3), (24,4), (25,4)}; fill((0,0)--(0,5)--(1,5)--(1,0)--cycle,pink); fill((1,0)--(1,3)--(2,3)--(2,0)--cycle,green); fill((2,0)--(2,2)--(3,2)--(3,0)--cycle,yellow); fill((3,4)--(3,5)--(4,5)--(4,4)--cycle,red); fill((3,0)--(3,4)--(4,4)--(4,0)--cycle,cyan); fill((11,0)--(11,5)--(12,5)--(12,0)--cycle,pink); fill((12,1)--(12,5)--(13,5)--(13,1)--cycle,cyan); fill((13,2)--(13,5)--(14,5)--(14,2)--cycle,green); fill((14,3)--(14,5)--(15,5)--(15,3)--cycle,yellow); fill((15,4)--(15,5)--(16,5)--(16,4)--cycle,red); fill((17,0)--(17,5)--(18,5)--(18,0)--cycle,pink); fill((18,0)--(18,4)--(19,4)--(19,0)--cycle,cyan); fill((18,4)--(18,5)--(19,5)--(19,4)--cycle,red); fill((19,0)--(19,3)--(20,3)--(20,0)--cycle,green); fill((19,3)--(19,5)--(20,5)--(20,3)--cycle,yellow); fill((21,4)--(21,5)--(22,5)--(22,4)--cycle,red); fill((22,1)--(22,5)--(23,5)--(23,1)--cycle,cyan); fill((23,0)--(23,5)--(24,5)--(24,0)--cycle,pink); fill((24,1)--(24,4)--(25,4)--(25,1)--cycle,green); fill((24,4)--(24,5)--(26,5)--(26,4)--cycle,yellow); int i; for(int i=0; i<15; i=i+1) { draw(shift(a[i])*p); draw(shift(b[i])*p); draw(shift(c[i])*p); draw(shift(d[i])*p); draw(shift(e[i])*p); } [/asy] \[\textbf{(A)}\qquad\qquad\qquad\textbf{(B)}\quad\qquad\qquad\textbf{(C)}\qquad\qquad\qquad\textbf{(D)}\quad\qquad\qquad\textbf{(E)}\] ~Basketball8 ~MRENTHUSIASM

Video Solution

https://youtu.be/USVVURBLaAc?t=171

See Also

2009 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
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

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: File missing

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