D. Absolute Sorting
constructive algorithms; math
You are given an array \(a\) consisting of \(n\) integers. The array is sorted if \(a_1 \le a_2 \le \dots \le a_n\).You want to make the array \(a\) sorted by applying the following operation exactly once: choose an integer \(x\), then for every \(i \in [1, n]\), replace \(a_i\) by \(|a_i - x|\). Find any value of \(x\) that will make the array sorted, or report that there is no such value.
The first line contains one integer \(t\) (\(1 \le t \le 2 \cdot 10^4\)) — the number of test cases.Each test case consists of two lines. The first line contains one integer \(n\) (\(2 \le n \le 2 \cdot 10^5\)). The second line contains \(n\) integers \(a_1, a_2, \dots, a_n\) (\(1 \le a_i \le 10^8\)).Additional constraint on the input: the sum of \(n\) over all test cases does not exceed \(2 \cdot 10^5\).
For each test case, print any integer \(x\) (\(0 \le x \le 10^9\)) that makes the array sorted. It can be shown that if such an integer \(x\) exists, there is at least one such integer between \(0\) and \(10^9\).If there is no such integer, then print \(-1\). If there are multiple suitable values of \(x\), print any of them.
In the first test case, after using \(x = 4\), the array becomes \([1, 1, 1, 1, 1]\).In the third test case, after using \(x = 0\), the array becomes \([1, 2, 3, 4, 5, 6, 7, 8]\).In the fourth test case, after using \(x = 42\), the array becomes \([32, 37, 38, 39, 40, 41]\).