640 math puzzles - 6 Mrs. Miniver’s Problem

Since the positions of the two circles are symmetrical about the central vertical line, we need only to consider one circle. Let the radius of the two circles be $1$ so that our requirement is to find $d$ , the separation between the two centers, such that $A_1 = A_2$ . Since $A_3$ is the reflection of $A_1$ , $A_1 = A_3$ , then $A_1=A_2=A_3 = A$ . We also note that together $A_1$ , $A_2$ , and $A_3$ form a circle. Therefore $A_1+A_2+A_3 = 3A = \pi \cdot 1^2 \implies A = \frac \pi 3$ .

To find $A$ , we consider $A_3$ . Let the central angle which extends the segment $A_3$ be $\theta$ . Then we have:

$A = A_3 = \underbrace{\frac \theta 2}_{\text{area of sector}} - \underbrace{\frac {\sin \theta}2}_{\text{area of triangle}} = \frac \pi 3$ $\implies 3\sin \theta + 2\pi = 3 \theta$

Solving the equation numerically, we have $\theta \approx 2.60533$ . And $d = 2 \cos \dfrac \theta 2 \approx 0.529859998$ , then $\left \lfloor 10000\dfrac dr \right \rfloor = \boxed{5298}$ .

If the lens (white) is to equal the two crescents (red), then the lens must be twice the area of one crescent. But since the lens plus the crescent equals one circle, we can say the lens must be $\frac{2}{3}$ of the circle.

The area of a lens is given by this formula :

$A = \pi{r^2} - 2r^2\tan^{-1}\left(\dfrac{d}{\sqrt{4r^2 - d^2}}\right) - \dfrac{{d}\sqrt{4r^2-d^2}}{2}$

Let $r = 1$ and equate to $\frac{2}{3}$ of a circle:

$\pi - 2\tan^{-1}\left(\dfrac{d}{\sqrt{4- d^2}}\right) - \dfrac{{d}\sqrt{4-d^2}}{2} = \dfrac{2\pi}{3}$

Solving numerically , $d\approx 0.529864$ , or $\Bigl\lfloor{10000\dfrac{d}{r}}\Bigr\rfloor = \boxed{5298}$

By phone calculator's iterations, I got d = 0.52986417 for r = 1.

Lens area (A) with distance d between two equal circles of unit radius is put equal as 2*pi-2A=A, which gives equation for d as

2Tan⁻¹[d/√(4-d²)]+(d/2)√(4-d²)=(⊓/3)

which is solved using WolframAlpha for distance d between two circles as 0.529864.

Answer = 5298

WolframAlpha find $d ≈ 0.529864169205554...$ for $r=1$ and answer is $5298$ .