What is the LCM of a and b if $(a+2 \sqrt{2} )/b$ is the ration of the area of larger circle and smaller circles?

Question

Radius of all four smaller circles is $R$. If the ratio between the area of the larger circle and the sum of areas of the smallest circles is $(a+2 \sqrt{2} )/b$ then find the LCM of $a$ and $b$.

Source: Bangladesh Math Olympiad 2014 Junior Category

I don't know how to get the ration between the area of the larger circle and sum of the area of the smaller circles. Can anyone help me with a hint or formula so that I can overcome this problem?

Answer 1

Centers of small circles form a square with side $2R$. Diameter of the big circle is $2R+2R\sqrt2$. Radius of the big circle is therefore $R(1+\sqrt2)$. Check jmerry’s picture.

Ratio of big circle area and small circles area is:

$$\frac{R^2(1+\sqrt2)^2\pi}{4R^2\pi}=\frac{3+2\sqrt2}{4}=\frac{a+2\sqrt2}{b}$$.

$$b=\frac{4a+8\sqrt2}{3+2\sqrt2}\cdot\frac{3-2\sqrt2}{3-2\sqrt2}$$

$$b=12a-32-\sqrt2(8a-24)$$

$b$ has to be integer and therefore $a=3, b=4.$

Answer 2

Draw in the centers and the segments between them:

Now, can you find the distance $OO_1$ between the center of the big circle and the center of one of the small circles, in terms of $R$? Can you get the radius of the big circle from that?