D. Split Plus K
greedy; math; number theory
eliteLAQ - Desert Ruins⠀There are \(n\) positive integers \(a_1, a_2, \dots, a_n\) on a blackboard. You are also given a positive integer \(k\). You can perform the following operation some (possibly \(0\)) times: choose a number \(x\) on the blackboard; erase one occurrence of \(x\); write two positive integers \(y\), \(z\) such that \(y+z = x+k\) on the blackboard. Is it possible to make all the numbers on the blackboard equal? If yes, what is the minimum number of operations you need?
Each test contains multiple test cases. The first line contains the number of test cases \(t\) (\(1 \le t \le 10^4\)). The description of the test cases follows.The first line of each test case contains two integers \(n\), \(k\) (\(1 \le n \le 2 \cdot 10^5\), \(1 \leq k \leq 10^{12}\)) — the number of integers initially on the blackboard and the constant \(k\).The second line of each test case contains \(n\) integers \(a_1, a_2, \ldots, a_n\) (\(1 \le a_i \le 10^{12}\)) — the initial state of the blackboard.It is guaranteed that the sum of \(n\) over all test cases does not exceed \(2 \cdot 10^5\).
For each test case, output a single line containing an integer: the minimum number of operations you need to make all the numbers on the blackboard equal, or \(-1\) if it is impossible.
In the first test case, \(k = 1\). You can make all the numbers on the blackboard equal to \(2\) with the following operations: Erase \(x = 4\) and write \((y, z) = (2, 3)\). Note that \(y+z=x+k\). The blackboard now contains the multiset \(\{3, 2, 3\}\). Erase \(x = 3\) and write \((y, z) = (2, 2)\). Note that \(y+z=x+k\). The blackboard now contains \(\{2, 2, 2, 3\}\). Erase \(x = 3\) and write \((y, z) = (2, 2)\). Note that \(y+z=x+k\). The blackboard now contains \(\{2, 2, 2, 2, 2\}\). This makes all the numbers equal in \(3\) operations. It can be shown that you cannot make all the numbers equal in less than \(3\) operations.In the second test case, \(k = 3\). You can make all the numbers on the blackboard equal to \(7\) with the following operation: Erase \(x = 11\) and write \((y, z) = (7, 7)\). Note that \(y+z=x+k\). The blackboard now contains \(\{7, 7, 7\}\). In the third test case, \(k = 10\). You can make all the numbers on the blackboard equal to \(40\) with the following operations: Erase \(x = 100\) and write \((y, z) = (70, 40)\). Note that \(y+z=x+k\). The blackboard now contains \(\{70, 40, 40, 100\}\). Erase \(x = 70\) and write \((y, z) = (40, 40)\). Note that \(y+z=x+k\). The blackboard now contains \(\{40, 40, 40, 40, 100\}\). Erase \(x = 100\) and write \((y, z) = (40, 70)\). Note that \(y+z=x+k\). The blackboard now contains \(\{40, 40, 40, 40, 40, 70\}\). Erase \(x = 70\) and write \((y, z) = (40, 40)\). Note that \(y+z=x+k\). The blackboard now contains \(\{40, 40, 40, 40, 40, 40, 40\}\). In the fourth and in the fifth test case, you can show that it is impossible to make all the numbers on the blackboard equal.