A unit circle is cut into 2012 pieces (all equal in arc length) by 2012 distinct points. Three of them are chosen at random. The probability that these three points form a right triangle is m/n where m and n are relatively prime integers. What is m + n?

Question

ganeshie8 · Answer

2012

ganeshie8 · Answer

m/n = 1/2011

ganeshie8 · Answer

angle in a semi-circle is a right angle

anonymous · Answer

Wait, isn't there more probablility since each possible diameter can have 1004 possible right triangles?

ganeshie8 · Answer

two of the three points must form a diameter

anonymous · Answer

Yes. Knowing that though, it can't possibly be 2011 possible triangle...

anonymous · Answer

Oh...turns out I was on the right track...
$$\frac{2010 \times 1006}{\left(\begin{matrix}2012 \ 3\end{matrix}\right)}$$

ganeshie8 · Answer

yeah... but that comes to be very huge number... to simlify

ganeshie8 · Answer

*to simplify

anonymous · Answer

$$\frac{2010 \times 1006}{\frac{2012 \times 2011 \times 2010}{3 \times 2}} \implies \frac{2010 \times 1006}{1006 \times 2011 \times 2010} \times 3 \implies \frac{3}{2011}$$Does this work? Then m + n = 2014?

ganeshie8 · Answer

wow looks correct to me also :)

anonymous · Answer

ALright. Thanks :)

ganeshie8 · Answer

:)