12) A gold coin is surrounded by four equal silver coins. Each silver coin is tangent to the gold coin and to two other silver coins. Find
the ratio between the radius of the gold coin and that of the silver coins.

![](https://cdn.mathpix.com/cropped/2024_04_17_cd047c2caa32abeb5404g-2.jpg?height=63&width=772&top_left_y=155&top_left_x=181)
12. The answer is $\mathbf{( B )}$.

Let $r$ and $R$ be the radii of the gold coin and the silver coins, respectively. Referring to the figure below, where the five coins are drawn, we observe that the triangle $A B C$ is a right-angled (at vertex $C$) and isosceles triangle. Moreover, the hypotenuse $A B$ has a length of $2(R+r)$, while the legs $A C$ and $B C$ have a length of $2 R$.

![](https://cdn.mathpix.com/cropped/2024_04_17_e415f128e14398de33bcg-4.jpg?height=425&width=434&top_left_y=1027&top_left_x=865)

Thus, $2(R+r)=2 \sqrt{2} R$, which simplifies to $R+r=\sqrt{2} R$. Dividing both sides by $R$ gives

$$
1+\frac{r}{R}=\sqrt{2} \text {. }
$$

From this, it follows that $\frac{r}{R}=\sqrt{2}-1$.

[Problem proposed by K. Kuzmin.]