Problem 620
Planetary Gears
A circle $C$ of circumference $c$ centimetres has a smaller circle $S$ of circumference $s$ centimetres lying off-centre within it. Four other distinct circles, which we call “planets”, with circumferences $p$, $p$, $q$, $q$ centimetres respectively ($p<q$), are inscribed within $C$ but outside $S$, with each planet touching both $C$ and $S$ tangentially. The planets are permitted to overlap one another, but the boundaries of $S$ and $C$ must be at least 1cm apart at their closest point.
Now suppose that these circles are actually gears with perfectly meshing teeth at a pitch of 1cm. $C$ is an internal gear with teeth on the inside. We require that $c$, $s$, $p$, $q$ are all integers (as they are the numbers of teeth), and we further stipulate that any gear must have at least 5 teeth.
Note that “perfectly meshing” means that as the gears rotate, the ratio between their angular velocities remains constant, and the teeth of one gear perfectly align with the groves of the other gear and vice versa. Only for certain gear sizes and positions will it be possible for $S$ and $C$ each to mesh perfectly with all the planets. Arrangements where not all gears mesh perfectly are not valid.
Define $g(c,s,p,q)$ to be the number of such gear arrangements for given values of $c$, $s$, $p$, $q$: it turns out that this is finite as only certain discrete arrangements are possible satisfying the above conditions. For example, $g(16,5,5,6)=9$.
Here is one such arrangement:
Let $G(n)=\sum_{s+p+q\le n}g(s+p+q,s,p,q)$ where the sum only includes cases with $p<q$, $p\ge5$, and $s\ge5$, all integers. You are given that $G(16)=9$ and $G(20)=205$.
Find $G(500)$.
行星齿轮
在周长为$c$厘米的圆$C$内部有一个不同心的周长为$s$厘米的小圆$S$。在$C$的内部但在$S$的外部，还有四个不同的圆，我们称之为“行星”，其周长分别是$p$，$p$，$q$，$q$厘米（$p<q$），与$C$和$S$均相切。行星之间允许相互重叠，但是$S$和$C$的圆周之间最接近的地方至少要相距1厘米。
现在假设这些圆其实是一组节距1厘米且完美咬合的齿轮，其中$C$是齿位于内部的内齿轮。我们要求$c$，$s$，$p$，$q$均为整数（因而它们代表齿的数量），并进一步要求所有齿轮至少要有5个齿。
注意“完美咬合”的意思是当齿轮转动时，它们的角速度之比为常数，而且任一齿轮的齿都完美对齐另一齿轮的槽，反之亦然。只有特定的齿轮尺寸和位置能够让$S$和$C$与所有行星完美咬合。只有所有齿轮均完美咬合的设计方式才视为有效。
记$g(c,s,p,q)$为给定$c$，$s$，$p$，$q$时上述有效齿轮设计方式的数量：这一数量总是有限的，因为只有一部分离散的设计方式可能满足上述条件。例如，$g(16,5,5,6)=9$。
下图展示了一种有效的设计方式:
记$G(n)=\sum_{s+p+q\le n}g(s+p+q,s,p,q)$，其中被求和的参数还需满足$p<q$，$p\ge5$和$s\ge5$，且均为整数。已知$G(16)=9$以及$G(20)=205$。
求$G(500)$。