Difference between revisions of "2009 AMC 8 Problems/Problem 4"
Mrdavid445 (talk | contribs) m |
MRENTHUSIASM (talk | contribs) (Deleted repetitive solution.) |
||
(21 intermediate revisions by 10 users not shown) | |||
Line 1: | Line 1: | ||
==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}} |
Revision as of 00:56, 26 November 2023
Contents
[hide]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.