C. XOR and Triangle
bitmasks; brute force; geometry; greedy; probabilities
This time, the pink soldiers have given you an integer \(x\) (\(x \ge 2\)).Please determine if there exists a positive integer \(y\) that satisfies the following conditions. \(y\) is strictly less than \(x\). There exists a non-degenerate triangle\(^{\text{∗}}\) with side lengths \(x\), \(y\), \(x \oplus y\). Here, \(\oplus\) denotes the bitwise XOR operation. Additionally, if there exists such an integer \(y\), output any.\(^{\text{∗}}\)A triangle with side lengths \(a\), \(b\), \(c\) is non-degenerate when \(a+b > c\), \(a+c > b\), \(b+c > a\).
Each test contains multiple test cases. The first line contains the number of test cases \(t\) (\(1 \le t \le 2000\)). The description of the test cases follows. The only line of each test case contains a single integer \(x\) (\(2 \le x \le 10^9\)).
For each test case, print one integer on a separate line. The integer you must output is as follows: If there exists an integer \(y\) satisfying the conditions, output the value of \(y\) (\(1 \le y < x\)); Otherwise, output \(-1\). If there exist multiple integers that satisfy the conditions, you may output any.
In the first test case, there exists a non-degenerate triangle with side lengths \(3\), \(5\), and \(3 \oplus 5 = 6\). Therefore, \(y=3\) is a valid answer.In the second test case, \(1\) is the only possible candidate for \(y\), but it cannot make a non-degenerate triangle. Therefore, the answer is \(-1\).