550 「LibreOJ Round #8」Matching

Given a grid with $n$ rows and $m$ columns, in this grid, you can choose two different grid points and match them into a pair. Notice that a point can only be matched with no more than one point. It means unmatched points are allowed. We define a weight function $f(A, B)$ for a pair of points $A, B$ as the following form: $$\begin{aligned} dx &= \left|x_A - x_B\right| \\ dy &= \left|y_A - y_B\right| \\ f(A, B) &= \begin{cases} dx + dy & , dx + dy \geq k \\ 0 & , dx + dy < k \end{cases} \end{aligned}$$ The weight of a possible matching scheme is defined as the sum of weights that the matching pairs given. Your task is to calculate the maximum possible weight of all matching schemes.

Read from the standard input.

The first line contains a single integer $T$ which means the number of the test cases.

Each of the following $T$ lines contains three integers $n, m, k$ which means the height and the width of the grid, and the constant $k$ in the weight function.

Write to the standard output.

For each test case, output a line containing a single integer — the maximum of the weight of possible matching plans.

As for $1 \times 1$ grid, there is no point for the only one to match, so the answer is $0$.

As for $1 \times 2$ grid, we match the only two points into a pair, so the answer is $1$.

As for $2 \times 2$ grid, we match the top left point and the bottom right one into a pair, the top right point and the bottom left one into a pair. The answer is $4$, as shown by the following picture.

As for $2 \times 3$ grid, we match the top left point and the bottom middle one, the top middle point and the bottom right one, the top right point and the bottom left one. The answer is $7$, as shown by the following picture.

For all test cases, $1 \leq T \leq 10^5$，$1 \leq n,m \leq 10^6$，$0 \leq k \leq \min(n, m) - 1$.

Detailed constraints are as follows (blank grids denote the same constraints as mentioned above):

Subtask#	Score (percentage)	$T$	$n, m$	Special Constraints
$1$	$10$	$\leq 20$	$\leq 2000$	$n \leq 2$
$2$	$15$	$\leq 5$	$\leq 8$	$n \times m \leq 10$ and $k = \min(n, m) - 1$
$3$	$25$	$\leq 3$	$\leq 16$	-
$4$	$18$	$\leq 100$	$\leq 5000$	$n = m$ and $n \equiv 0 \pmod 2$
$5$	$32$	$\leq 10^5$	$\leq 10^6$	-