Let a, b, c, d, e, f, g, h be distinct elements in the set \{-7, -5, -3, -2, 2, 4, 6, 13\}. The minimum value of ( a + b + c + d ) 2 + ( e + f + g + h ) 2 is

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Note that the sum of the elements is 8. Let $a + b + c + d = x$ .

Therefore, $e + f + g + h = 8 - x$ .

Again, let $y = x^{2} + (8 - x)^{2}$ .

Therefore, $y = 2 x^{2} - 16 x + 64 = 2 [x^{2} - 8 x + 32] = 2 [(x - 4)^{2} + 16]$ .

The minimum value is $32$ when $x = 4$ .

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