AMC10 2022 B

AMC10 2022 B · Q18

AMC10 2022 B · Q18. It mainly tests Systems of equations, Basic counting (rules of product/sum).

Consider systems of three linear equations with unknowns $x$, $y$, and $z$, \begin{align*} 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{align*} 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.)

考虑含有未知数 $x$、$y$ 和 $z$ 的三个线性方程组， \begin{align*} 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{align*} 其中每个系数要么是 $0$ 要么是 $1$，且该方程组有非平凡解（即非 $x=y=z=0$ 的解）。例如，一个这样的方程组是 \[\{ 1x + 1y + 0z = 0, 0x + 1y + 1z = 0, 0x + 0y + 0z = 0 \}\]\n它有一个非零解 $\{x,y,z\} = \{1, -1, 1\}$。有多少这样的方程组？（方程组中的方程不必互异，且含有相同方程但顺序不同的两个方程组被视为不同。）

(A) 302 302

(B) 338 338

(D) 343 343

(E) 344 344

Answer

Correct choice: (B)

正确答案：（B）

Solution

Let $M_1=\begin{bmatrix}a_1 & b_1 & c_1\end{bmatrix}, M_2=\begin{bmatrix}a_2 & b_2 & c_2\end{bmatrix},$ and $M_3=\begin{bmatrix}a_3 & b_3 & c_3\end{bmatrix}.$ We wish to count the ordered triples $(M_1,M_2,M_3)$ of row matrices. We perform casework: 1. $M_1=M_2=M_3.$ 2. Exactly two of $M_1,M_2,$ and $M_3$ are equal. 3. All of $M_1,M_2,$ and $M_3$ are different. Together, the answer is $8+168+126+36=\boxed{\textbf{(B)}\ 338}.$

令 $M_1=\begin{bmatrix}a_1 & b_1 & c_1\end{bmatrix}, M_2=\begin{bmatrix}a_2 & b_2 & c_2\end{bmatrix},$ 和 $M_3=\begin{bmatrix}a_3 & b_3 & c_3\end{bmatrix}$。我们希望计数有序三元组 $(M_1,M_2,M_3)$ 的行矩阵。我们进行分类讨论： 1. $M_1=M_2=M_3$。 2. 恰好有两个 $M_1,M_2,$ 和 $M_3$ 相等。 3. $M_1,M_2,$ 和 $M_3$ 均不同。总计，答案是 $8+168+126+36=\boxed{\textbf{(B)}\ 338}$。

Topics