$3 \cdot 37$ Three meshing gears $A, B, C$ have the number of teeth $x, y, z$ (as shown in the figure, all gears have the same tooth size and are evenly spaced).
The ratio of the angular velocities of $A, B, C$ per minute is
(A) $x: y: z$.
(B) $z: y: x$.
(C) $y: z: x$.
(D) $y z: x z: x y$.
(E) $x z: y x: z y$.
(31st American High School Mathematics Examination, 1980)
[Solution] Since the three gears have the same tooth size, are evenly spaced, and mesh with each other, they all run at the same absolute speed $v$ ( $v$ is the distance a point on the circumference moves in a unit of time). Let $\alpha, \beta, \gamma$ be the angular velocities of $A, B, C$ respectively. If $a, b, c$ are the circumferences of $A, B, C$ respectively, then
$$
\alpha=\frac{v}{a}, \quad \beta=\frac{v}{b}, \quad \gamma=\frac{v}{c} .
$$

Therefore,
$$
\alpha a=\beta b=\gamma c,
$$

or
$$
\frac{\alpha}{\frac{1}{a}}=\frac{\beta}{\frac{1}{b}}=-\frac{\gamma}{\frac{1}{c}} .
$$

Thus, the ratio of their angular velocities is $\frac{1}{a}: \frac{1}{b}: \frac{1}{c}$. Since $a, b, c$ are proportional to $x, y, z$, the ratio of the angular velocities is $\frac{1}{x}: \frac{1}{y}: \frac{1}{z}$. That is,
$$
y z: x z: x y .
$$

Therefore, the answer is $(D)$.