Question

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 $\begin{aligned} &(a + b + c + d)^2 \\ &\quad {}+ (e + f + g + h)^2? \end{aligned}$

$30$

$32$

$34$

$40$

$50$

Answer: C

Concepts:optimization completing the square

Difficulty rating: 1910

Solution:

The elements sum to $8.$ If $a + b + c + d = x,$ then $e + f + g + h = 8 - x,$ so $x^2 + (8 - x)^2 = 2(x - 4)^2 + 32.$

This is minimized when $x = 4,$ giving $32.$ But $13$ must lie in one group, and no three of the remaining elements add with $13$ to make $4$ (that would need three of them to sum to $-9$ ). With $-7,$ the other two would need to sum to $-2,$ which no available pair does; without $-7,$ replacing any term in $-5-3-2=-10$ raises the sum past $-9.$ So $x = 4$ is unattainable and $(x - 4)^2 \ge 1.$

The minimum is $2(1) + 32 = 34,$ achieved for instance by $\{-7, -5, 2, 13\}$ (sum $3$ ) and $\{-3, -2, 4, 6\}$ (sum $5$ ).

Thus, the correct answer is C.

2005 AMC 12B Problem 20

Answer: C

Difficulty rating: 1910

Solution:

Problem 20 in Other Years