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.

We are given a set \( S = \{-7, -5, -3, -2, 2, 4, 6, 13\} \) and tasked with finding the minimum possible value of the expression

\[
(a + b + c + d)^2 + (e + f + g + h)^2
\]

where \(a, b, c, d, e, f, g, h\) are distinct elements from \(S\).

### Step 1: Partition the set
We are splitting the set into two groups, each with four elements. Let:
- \(A = \{a, b, c, d\}\)
- \(B = \{e, f, g, h\}\)

The goal is to minimize \( (S_A)^2 + (S_B)^2 \), where \( S_A = a + b + c + d \) and \( S_B = e + f + g + h \), subject to \( S_A + S_B = S_{\text{total}} \).

The total sum of all elements in \(S\) is:

\[
S_{\text{total}} = -7 + (-5) + (-3) + (-2) + 2 + 4 + 6 + 13 = 8
\]

Thus, we want to minimize \( (S_A)^2 + (S_B)^2 \) with the constraint that \( S_A + S_B = 8 \). Since \( S_B = 8 - S_A \), the expression becomes:

\[
(S_A)^2 + (8 - S_A)^2
\]

### Step 2: Minimize the expression
We now seek to minimize the function:

\[
f(S_A) = (S_A)^2 + (8 - S_A)^2
\]

Expanding this:

\[
f(S_A) = S_A^2 + (64 - 16S_A + S_A^2) = 2S_A^2 - 16S_A + 64
\]

To find the minimum, take the derivative and set it equal to zero:

\[
\frac{d}{dS_A} (2S_A^2 - 16S_A + 64) = 4S_A - 16
\]

Setting \( 4S_A - 16 = 0 \), we get:

\[
S_A = 4
\]

Thus, the minimum occurs when \( S_A = 4 \), and therefore \( S_B = 8 - 4 = 4 \).

### Step 3: Verify the partition
We need to find two groups of four numbers from the set \( S \) such that the sum of the numbers in each group is 4. One possible partition is:

- Group A: \( \{-7, 13, -2, 0\} \) (which sums to \( 4 \))
- Group B: \( \{-5, -3, 6, 2\} \) (which also sums to  4)

### Step 4: Calculate the minimum value
Using the partition \(S_A = 455%\&guaranteenmaximum %3append

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?

Learn how to find the minimum possible value of the expression (a + b + c + d)^2 + (e + f + g + h)^2 with distinct elements from the set {–7, –5, –3, –2, 2, 4, 6, 13}. This problem involves quadratic functions, optimization, and combinatorial partitioning techniques, suitable for students in Grades 11-12.

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