Difference between revisions of "2003 AMC 12A Problems/Problem 12"

Revision as of 10:13, 11 November 2006

Problem

Sally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6$ . She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards?

$\mathrm{(A) \ } 8\qquad \mathrm{(B) \ } 9\qquad \mathrm{(C) \ } 10\qquad \mathrm{(D) \ } 11\qquad \mathrm{(E) \ } 12$

Solution

Let $R_i$ and $B_j$ designate the red card numbered $i$ and the blue card numbered $j$ , respectively.

$B_5$ is the only blue card that $R_5$ evenly divides, so $R_5$ must be at one end of the stack and $B_5$ must be the card next to it.

$R_1$ is the only other red card that evenly divides $B_5$ , so $R_1$ must be the other card next to $B_5$ .

$B_4$ is the only blue card that $R_4$ evenly divides, so $R_4$ must be at one end of the stack and $B_4$ must be the card next to it.

$R_2$ is the only other red card that evenly divides $B_4$ , so $R_2$ must be the other card next to $B_4$ .

$R_2$ doesn't evenly divide $B_3$ , so $B_3$ must be next to $R_1$ , $B_6$ must be next to $R_2$ , and $R_3$ must be in the middle.

This yeilds the following arrangement from top to bottom: $\{R_5,B_5,R_1,B_3,R_3,B_6,R_2,B_4,R_4\}$

Therefore, the sum of the numbers on the middle three cards is $3+3+6=12 \Rightarrow E$ .

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Difference between revisions of "2003 AMC 12A Problems/Problem 12"

Revision as of 10:13, 11 November 2006

Problem

Solution

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Revision as of 22:46, 9 November 2006 (view source) Xantos C. Guin (talk \| contribs) (added problem and solution)	Revision as of 10:13, 11 November 2006 (view source) JBL (talk \| contribs) m (2003 AMC 12A/Problem 12 moved to 2003 AMC 12A Problems/Problem 12) Newer edit →
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