A. Meaning Mean
data structures; greedy; math; sortings
Pak Chanek has an array \(a\) of \(n\) positive integers. Since he is currently learning how to calculate the floored average of two numbers, he wants to practice it on his array \(a\).While the array \(a\) has at least two elements, Pak Chanek will perform the following three-step operation: Pick two different indices \(i\) and \(j\) (\(1 \leq i, j \leq |a|\); \(i \neq j\)), note that \(|a|\) denotes the current size of the array \(a\). Append \(\lfloor \frac{a_i+a_j}{2} \rfloor\)\(^{\text{∗}}\) to the end of the array. Remove elements \(a_i\) and \(a_j\) from the array and concatenate the remaining parts of the array. For example, suppose that \(a=[5,4,3,2,1,1]\). If we choose \(i=1\) and \(j=5\), the resulting array will be \(a=[4,3,2,1,3]\). If we choose \(i=4\) and \(j=3\), the resulting array will be \(a=[5,4,1,1,2]\).After all operations, the array will consist of a single element \(x\). Find the maximum possible value of \(x\) if Pak Chanek performs the operations optimally.\(^{\text{∗}}\)\(\lfloor x \rfloor\) denotes the floor function of \(x\), which is the greatest integer that is less than or equal to \(x\). For example, \(\lfloor 6 \rfloor = 6\), \(\lfloor 2.5 \rfloor=2\), \(\lfloor -3.6 \rfloor=-4\) and \(\lfloor \pi \rfloor=3\)
Each test contains multiple test cases. The first line contains the number of test cases \(t\) (\(1 \le t \le 5000\)). The description of the test cases follows.The first line of each test case contains a single integer \(n\) (\(2 \le n \le 50\)) — the length of the array \(a\).The second line of each test case contains \(n\) integers \(a_1, a_2, \ldots, a_n\) (\(1 \le a_i \le 10^9\)) — the elements of the array \(a\).Do note that the sum of \(n\) over all test cases is not bounded.
For each test case, output a single integer: the maximum possible value of \(x\) after all numbers have been picked.
In the first test case, the array is initially \(a=[1,7,8,4,5]\). Pak Chanek will perform the following operations: Pick \(i=1\) and \(j=2\), then \(a=[8,4,5,4]\). Pick \(i=3\) and \(j=2\), then \(a=[8,4,4]\). Pick \(i=2\) and \(j=3\), then \(a=[8,4]\). Pick \(i=1\) and \(j=2\), then \(a=[6]\). After all the operations, the array consists of a single element \(x=6\). It can be proven that there is no series of operations that results in \(x\) greater than \(6\) in the end.