CF1637D Yet Another Minimization Problem

You are given two arrays $ a $ and $ b $ , both of length $ n $ . You can perform the following operation any number of times (possibly zero): select an index $ i $ ( $ 1 \leq i \leq n $ ) and swap $ a_i $ and $ b_i $ . Let's define the cost of the array $ a $ as $ \sum_{i=1}^{n} \sum_{j=i + 1}^{n} (a_i + a_j)^2 $ . Similarly, the cost of the array $ b $ is $ \sum_{i=1}^{n} \sum_{j=i + 1}^{n} (b_i + b_j)^2 $ . Your task is to minimize the total cost of two arrays.

Each test case consists of several test cases. The first line contains a single integer $ t $ ( $ 1 \leq t \leq 40 $ ) — the number of test cases. The following is a description of the input data sets. The first line of each test case contains an integer $ n $ ( $ 1 \leq n \leq 100 $ ) — the length of both arrays. The second line of each test case contains $ n $ integers $ a_1, a_2, \ldots, a_n $ ( $ 1 \leq a_i \leq 100 $ ) — elements of the first array. The third line of each test case contains $ n $ integers $ b_1, b_2, \ldots, b_n $ ( $ 1 \leq b_i \leq 100 $ ) — elements of the second array. It is guaranteed that the sum of $ n $ over all test cases does not exceed $ 100 $ .

For each test case, print the minimum possible total cost.

In the second test case, in one of the optimal answers after all operations $ a = [2, 6, 4, 6] $ , $ b = [3, 7, 6, 1] $ . The cost of the array $ a $ equals to $ (2 + 6)^2 + (2 + 4)^2 + (2 + 6)^2 + (6 + 4)^2 + (6 + 6)^2 + (4 + 6)^2 = 508 $ . The cost of the array $ b $ equals to $ (3 + 7)^2 + (3 + 6)^2 + (3 + 1)^2 + (7 + 6)^2 + (7 + 1)^2 + (6 + 1)^2 = 479 $ . The total cost of two arrays equals to $ 508 + 479 = 987 $ .