Circus

Geometry Level 5

The Red and Blue circles share the same center.

The radius of Red circle equals 1 1 + 1 2 + 1 1 + 1 2 + 1 1 + ⋱ \cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{1 + \ddots}}}}} and the radius of Blue circle equals 3 − 1 4 + 1 ⋅ 3 4 ⋅ 6 − 1 ⋅ 3 ⋅ 5 4 ⋅ 6 ⋅ 8 + ⋯ 3 - \frac{1}{4} + \frac{1\cdot3}{4\cdot6} - \frac{1\cdot3\cdot5}{4\cdot6\cdot8} + \cdots If the hatched area equals ( L M N − 1 ) π + M \bigg(\frac{L}{M\sqrt{N}}-1\bigg)\pi+M where L , M , N L,M,N are positive co-prime integers with N N being square-free. Find L + M + N L+M+N .


The answer is 12.

Digvijay Singh by Digvijay Singh , 21, India

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1 solution

Manuel Kahayon
Apr 3, 2016

RAAAAR took me more than 10 minutes TT.TT

Anyways, you will need:

Construction skills

Binomial Expansion Theorem

Continued Fractions

Compound figures

s i n ( 15 ) = 6 − 2 4 sin (15) = \frac{\sqrt{6}-\sqrt{2}}{4}

And you get the answer!

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Could u please explain how??

Akash Mishra - 5 years, 2 months ago

Please add more details to the solution so that others can undertand what you are trying to say.

Calvin Lin Staff - 5 years, 2 months ago

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