2005 AMC 12B Problems/Problem 20

Problem

Let $a,b,c,d,e,f,g$ and $h$ be distinct elements in the set $\{-7,-5,-3,-2,2,4,6,13\}.$

What is the minimum possible value of $(a+b+c+d)^{2}+(e+f+g+h)^{2}?$

$\mathrm{(A)}\ 30 \qquad \mathrm{(B)}\ 32 \qquad \mathrm{(C)}\ 34 \qquad \mathrm{(D)}\ 40 \qquad \mathrm{(E)}\ 50$

Solution

The sum of the set is $-7-5-3-2+2+4+6+13=8$ , so if we could have the sum in each set of parenthesis be $4$ then the minimum value would be $2(4^2)=32$ . Considering the set of four terms containing $13$ , this sum could only be even if it had two or four odd terms. If it had all four odd terms then it would be $13-7-5-3=-2$ , and with two odd terms then its minimum value is $13-7+2-2=6$ , so we cannot achieve two sums of $4$ . The closest we could have to $4$ and $4$ is $3$ and $5$ , which can be achieved through $13-7-5+2$ and $6-3-2+4$ . So the minimum possible value is $3^2+5^2=34\Rightarrow\boxed{C}$ .

2005 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
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2005 AMC 12B Problems/Problem 20

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