an inversion week

2017 Multi-University Training Contests went on on Tuesday and Thursday. As usual, Fizzydavid carried our team. In Round 3, our team got too much penalty time, for I messed up problem 1004. Then I tried to solve problem 1002, but failed because of a lack of math knowledge. It was a pretty beautiful problem. Here's the problem. You're given two arrays \(A[0...2^m-1]\) and \(B[0...2^m-1]\). You need to calculate the array \(C[0...2^m-1]\) where \(C[k]\ =\ \sum_{i\ and\ j\ =\ k}\ \ \ A[i\ xor\ j]\ \times\ B[i\ or\ j]\).\((1\ \leq\ m\ \leq\ 19)\). Shanquan2 solved it in just half an hour. Can you solve it as fast as he did? Problems in round 4 were easier than before, and there's nothing more to talk about them.

ARC080 was held on Sunday evening. My performance was totally a flop. I spent more than one hour on problem C. However, I was wrong at the first step without noticing that. As a result, my rank dropped to 20 at the end of the contest.

Last week, I've mentioned CF833C. The trick is that the answer is at most \(C_{27}^{18}\). So all you need to do is to enumerate every possible \(f(x)\) value and check whether it's in the range.