2013 AMC 12B Problems/Problem 12: Difference between revisions
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Revision as of 13:03, 3 July 2013
Problem 12
Cities 
, 
, 
, 
, and 
are connected by roads 
, 
, 
, 
, 
, 
, and 
. How many different routes are there from to that use each road exactly once? (Such a route will necessarily visit some cities more than once.)
![[asy] unitsize(10mm); defaultpen(linewidth(1.2pt)+fontsize(10pt)); dotfactor=4; pair A=(1,0), B=(4.24,0), C=(5.24,3.08), D=(2.62,4.98), E=(0,3.08); dot (A); dot (B); dot (C); dot (D); dot (E); label("$A$",A,S); label("$B$",B,SE); label("$C$",C,E); label("$D$",D,N); label("$E$",E,W); guide squiggly(path g, real stepsize, real slope=45) { real len = arclength(g); real step = len / round(len / stepsize); guide squig; for (real u = 0; u < len; u += step){ real a = arctime(g, u); real b = arctime(g, u + step / 2); pair p = point(g, a); pair q = point(g, b); pair np = unit( rotate(slope) * dir(g,a)); pair nq = unit( rotate(0 - slope) * dir(g,b)); squig = squig .. p{np} .. q{nq}; } squig = squig .. point(g, length(g)){unit(rotate(slope)*dir(g,length(g)))}; return squig; } pen pp = defaultpen + 2.718; draw(squiggly(A--B, 4.04, 30), pp); draw(squiggly(A--D, 7.777, 20), pp); draw(squiggly(A--E, 5.050, 15), pp); draw(squiggly(B--C, 5.050, 15), pp); draw(squiggly(B--D, 4.04, 20), pp); draw(squiggly(C--D, 2.718, 20), pp); draw(squiggly(D--E, 2.718, -60), pp);[/asy]](http://latex.artofproblemsolving.com/c/a/c/cac293a82b8e86665148f7b48284aa26ff4a80c8.png)
Solution
Note that cities and can be removed when counting paths because if a path goes in to or , there is only one possible path to take out of cities or .
So the diagram is as follows:
![[asy] unitsize(10mm); defaultpen(linewidth(1.2pt)+fontsize(10pt)); dotfactor=4; pair A=(1,0), B=(4.24,0), C=(5.24,3.08), D=(2.62,4.98), E=(0,3.08); dot (A); dot (B); dot (D); label("$A$",A,S); label("$B$",B,SE); label("$D$",D,N); draw(A--B..D..cycle); draw(A--D); draw(B--D);[/asy]](http://latex.artofproblemsolving.com/2/e/c/2ec1905dd2842beab7a5c2246315eb415dde1560.png)
Now we proceed with casework. Remember that there are two ways to travel from to , to , to and to .:
Case 1 
: From , if the path returns to , then the next path must go to 
. There are 
possibilities of the path 
. If the path goes to from , then the path must continue with either 
or 
. There are 
possibilities. So, this case gives 
different possibilities.
Case 2 
: The path must continue with 
. There are 
possibilities for this case.
Putting the two cases together gives 
See also
| 2013 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 11 |
Followed by Problem 13 |
| 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 AMC 12 Problems and Solutions | |
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