{"status": "success", "data": {"description_md": "Consider systems of three linear equations with unknowns $x$, $y$, and $z$,\n\n$$\n\\begin{aligned}\na_1 x + b_1 y + c_1 z &= 0\\\\\na_2 x + b_2 y + c_2 z &= 0\\\\\na_3 x + b_3 y + c_3 z &= 0\n\\end{aligned}\n$$\n\nwhere each of the coefficients is either $0$ or $1$ and the system has a solution other than $x=y=z=0$.\n\nFor example, one such system is $$\\{ 1x + 1y + 0z = 0, 0x + 1y + 1z = 0, 0x + 0y + 0z = 0 \\}$$\n\nwith a nonzero solution of $\\{x,y,z\\} = \\{1, -1, 1\\}$. How many such systems of equations are there? (The equations in a system need not be distinct, and two systems containing the same equations in a different order are considered different.)\n\n$\\textbf{(A)}\\ 302 \\qquad\\textbf{(B)}\\ 338 \\qquad\\textbf{(C)}\\ 340 \\qquad\\textbf{(D)}\\ 343 \\qquad\\textbf{(E)}\\ 344$", "description_html": "

Consider systems of three linear equations with unknowns x, y, and z,

\\begin{aligned} a_1 x + b_1 y + c_1 z &= 0\\\\ a_2 x + b_2 y + c_2 z &= 0\\\\ a_3 x + b_3 y + c_3 z &= 0 \\end{aligned}

where each of the coefficients is either 0 or 1 and the system has a solution other than x=y=z=0.

For example, one such system is \\{ 1x + 1y + 0z = 0, 0x + 1y + 1z = 0, 0x + 0y + 0z = 0 \\}

with a nonzero solution of \\{x,y,z\\} = \\{1, -1, 1\\}. How many such systems of equations are there? (The equations in a system need not be distinct, and two systems containing the same equations in a different order are considered different.)

\\textbf{(A)}\\ 302 \\qquad\\textbf{(B)}\\ 338 \\qquad\\textbf{(C)}\\ 340 \\qquad\\textbf{(D)}\\ 343 \\qquad\\textbf{(E)}\\ 344

", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2022 AMC 10B Problem 18", "can_next": true, "can_prev": true, "nxt": "/problem/22_amc10B_p19", "prev": "/problem/22_amc10B_p17"}}