Circle buddies by themselves

Geometry Level 3

In the picture above, there are two circles, both with radius 1 1 . One circle passes through the other circle’s center (the red circle passes through the center of the blue circle, and vise versa).

The area of region I I , the region where the two circles intersect, can be expressed as:

a π + b c d \frac{a\pi +b\sqrt c}{d} ,

where a , b , c , d a,b,c,d are integers with c c square-free and d d minimized.

Evaluate 5 a + 5 b − 13 c + 6 d + c 2 5a +5b -13c +6d +c^2 .


The answer is 11.

Maximilian Million Kenney by Maximilian Million Kenney , 13

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1 solution

The area of the common region between the two circles is 4 π − 3 3 6 \dfrac {4π-3\sqrt 3}{6}

So, a = 4 , b = − 3 , c = 3 , d = 6 ⟹ 5 a + 5 b − 13 c + 6 d + c 2 = 20 − 15 − 39 + 36 + 9 = 11 a=4,b=-3,c=3,d=6\implies 5a+5b-13c+6d+c^2=20-15-39+36+9=\boxed {11} .

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