P14738 [ICPC 2021 Seoul R] Grid Triangle

A grid triangle in the 3-dimensional grid system is a triangle of three integral points including the origin $(0,0,0)$ that satisfy the following property: There exist three different positive integers $X, Y, Z$ such that for every pair of the three points of the triangle, you can rotate and translate the cuboid of size $X \times Y \times Z$ in parallel with the grid system so that the pair are diagonally opposite (and so the farthest way) vertices of the cuboid. For instance, the triangle of the three points $(0,0,0), (1,2,3), (-2,3,1)$ is a grid triangle with the cuboid of size $1 \times 2 \times 3$. More specifically, the two points $(1,2,3), (-2,3,1)$ are the diagonally opposite vertices of the cuboid $\{(x,y,z) \mid -2 \le x \le 1, 2 \le y \le 3, 1 \le z \le 3\}$ of size $3 \times 1 \times 2$; the two points $(0,0,0), (1,2,3)$ are the diagonally opposite vertices of the cuboid $\{(x,y,z) \mid 0 \le x \le 1, 0 \le y \le 2, 0 \le z \le 3\}$ of size $1 \times 2 \times 3$; and the two points $(0,0,0), (-2,3,1)$ are the diagonally opposite vertices of the cuboid $\{(x,y,z) \mid -2 \le x \le 0, 0 \le y \le 3, 0 \le z \le 1\}$ of size $2 \times 3 \times 1$. Further, all three cuboids are parallel with the grid system. Write a program to output the number of grid triangles within a bounded 3-dimensional grid system. The grid system is bounded by three given positive integers, $A, B, C$, in such a way that all points of grid triangles should be within $\{(x,y,z) \mid -A \le x \le A, -B \le y \le B, -C \le z \le C\}$.

Your program is to read from standard input. The input is exactly one line containing three integers, $A, B, C$ ($1 \le A, B, C \le 10,000,000$).

Your program is to write to standard output. Print exactly one line. The line should contain the number of grid triangles in the 3-dimensional grid system bounded by $A, B, C$.