3. Circles $k, l, m$ touch a common tangent at three different points and their centers lie on a line. Circles $k$ and $l$ as well as circles $l$ and $m$ have external tangency. Determine the radius of circle $l$, if the radii of circles $k$ and $m$ are $3 \mathrm{~cm}$ and $12 \mathrm{~cm}$.

The written part of the school round of category C takes place

on Thursday, January 26, 2006

so that it starts in the morning and the contestants have 4 hours of clean time to solve the problems. For each problem, a contestant can earn 6 points, and a successful solver is a student who scores 10 points or more. Allowed aids are writing and drawing supplies, school MF tables, and calculators without a graphical display. These details will be communicated to the students before the start of the competition.
3. The mutual position of the circles and their common tangents must look like in Fig. 2, where the points of tangency of the circles $k, l, m$ on the common tangent are denoted by $K, L, M$, and $U$,

![](https://cdn.mathpix.com/cropped/2024_04_17_ba17fecc41b82e1be23bg-3.jpg?height=652&width=1200&top_left_y=385&top_left_x=428)

Fig. 2

$V, W$ are their centers and $r$ is the radius of circle $l$ (in centimeters). From the right trapezoids $K L V U, L M W V, K M W U$, according to the Pythagorean theorem,

$$
\begin{aligned}
|K L|^{2} & =(r+3)^{2}-(r-3)^{2}=12 r \\
|L M|^{2} & =(12+r)^{2}-(12-r)^{2}=48 r
\end{aligned}
$$

and

$$
|K M|^{2}=(3+2 r+12)^{2}-(12-3)^{2}=4 r^{2}+60 r+144 .
$$

Since $|K L|+|L M|=|K M|$, we get from the first two relations

$$
|K M|^{2}=(|K L|+|L M|)^{2}=|K L|^{2}+2|K L||L M|+|L M|^{2}=60 r+2 \sqrt{12 \cdot 48} r
$$

which, together with the third relation, gives the equation for $r$

$$
4 r^{2}-48 r+144=0
$$

Since $4 r^{2}-48 r+144=4\left(r^{2}-12 r+36\right)=4(r-6)^{2}$, this equation has the unique solution $r=6$ and the radius of circle $l$ is therefore $6 \mathrm{~cm}$.

For a complete solution, award 6 points. For expressing the lengths of segments $K L, L M, K M$ in terms of $r$, award 3 points, and another 3 points for the correct calculation of the radius $r$.