Difference between revisions of "2009 AMC 8 Problems/Problem 4"
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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 | + | 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> | ||
+ | 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> | ||
+ | <cmath> \textbf{(A)}\qquad\qquad\qquad\textbf{(B)}\quad\qquad\qquad\textbf{(C)}\:\qquad\qquad\qquad\textbf{(D)}\quad\qquad\qquad\textbf{(E)} </cmath> | ||
+ | |||
+ | ==Solution== | ||
+ | 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> | <asy> | ||
Line 8: | Line 55: | ||
real r=0.5, s=1.5; | real r=0.5, s=1.5; | ||
path p=origin--(1,0)--(1,1)--(0,1)--cycle; | 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(p); | ||
draw(shift(s,r)*p); | draw(shift(s,r)*p); | ||
Line 22: | Line 84: | ||
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 33: | Line 96: | ||
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[] 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)}; | 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; | int i; | ||
Line 43: | Line 130: | ||
} | } | ||
</asy> | </asy> | ||
− | <cmath> \textbf{(A)}\qquad\qquad\qquad\textbf{(B)}\quad\qquad\qquad\textbf{(C)} | + | <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== | ||
+ | {{AMC8 box|year=2009|num-b=3|num-a=5}} | ||
+ | {{MAA Notice}} |
Latest revision as of 01:56, 26 November 2023
Contents
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?
Solution
The answer is because the longest piece cannot fit into the figure.
Note that the five pieces can be arranged to form the figures in and
as shown below:
~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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.