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

\n

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

\n

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.

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