Given that \(\omega_1 \) and \(\omega_2\) are two circles which intersects at points \(X\) and \(Y\). Let \(P \) be an arbitrary points on \(\omega_1\). Suppose that the lines \(PX\) and \(PY\) meet \(\omega_2\) again at points \(A\) and \(B\), respectively.

Prove that the circumcircles of all triangles \(PAB\) have the same radius.

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We can see that as we move the point \(P\) on the circumference of the circle\([\)excluding \(X\) and \(Y],\)the \(\angle XPY=\angle XP_1Y\) remains constant.So this shows that \(AB=A_1B_1.\)Now we use extended sin rule to complete the problem.

Let the circum-radius of \(\triangle PAB\) be \(R\) and \(\triangle P_1A_1B_1\) be \(R_1.\)

In \(\triangle PAB, \frac{AB}{sin\angle P}=2R\) and in \(\triangle P_1A_1B_1,\frac{A_1B_1}{sin\angle P_1}=\frac{AB}{sin\angle P}=2R_1.\) Therefore \(2R=2R_1\Rightarrow\boxed {R=R_1}.\)Hence Proved. – Ayush Rai · 2 months, 4 weeks ago

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– Priyanshu Mishra · 2 months, 3 weeks ago

Very good solution. Very much clear also. I understood it now.Log in to reply

– Ayush Rai · 2 months, 3 weeks ago

Thanks...My fav. is geometry.I see that u like olympiad problems.Can u give some links of ur problems?did u get selected in INMO??Log in to reply

I have that books in PDF.

Actually I am attending KVS INMO camp for INMO 2017. I have to give INMO directly.

By the way, from which book you do geometry problems? – Priyanshu Mishra · 2 months, 3 weeks ago

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My email id : harshus99@gmail.com – Harsh Shrivastava · 2 months, 3 weeks ago

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– Ayush Rai · 2 months, 3 weeks ago

and also give the INMO paper to this email.Did u write RMO this yearLog in to reply

– Priyanshu Mishra · 2 months, 3 weeks ago

Yes I appeared in RMO this year from KVS region.Log in to reply

– Ayush Rai · 2 months, 3 weeks ago

How many did u solve?Log in to reply

– Priyanshu Mishra · 2 months, 3 weeks ago

I solved 4 and half.Log in to reply

– Ayush Rai · 2 months, 3 weeks ago

Awesome! Send me the question paper also thenLog in to reply

I am currently out of Delhi. I will send you after 5 Nov. – Priyanshu Mishra · 2 months, 3 weeks ago

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– Ayush Rai · 2 months, 3 weeks ago

ok.no probs. Can u give ur email addressLog in to reply

– Priyanshu Mishra · 2 months, 3 weeks ago

priyanshu_2feb@rediffmail.comLog in to reply

– Ayush Rai · 2 months, 3 weeks ago

Sometimes co-exter or pre-college maths.Can u give the pdf of the books to this email : raiayush234@gmail.comLog in to reply

@Ayush Rai and @Harsh Shrivastava

I have sent IMO books and KVS RMO 2016 paper to your e-mails.

Check your mail. – Priyanshu Mishra · 2 months, 1 week ago

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– Ayush Rai · 2 months, 1 week ago

Thanks a lot bro.Log in to reply

@Harsh Shrivastava, @Ayush Rai,

I have 36 Olympiad books related to IMO. Do you want all of them? – Priyanshu Mishra · 2 months, 3 weeks ago

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– Harsh Shrivastava · 2 months, 3 weeks ago

Yeah I also want em all.Log in to reply

– Ayush Rai · 2 months, 3 weeks ago

yup.sure.send it to my email.Log in to reply

Can you tell me the author of co-exter book? – Priyanshu Mishra · 2 months, 3 weeks ago

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– Ayush Rai · 2 months, 3 weeks ago

H. S. M. Coxeter and S. L. GreitzerLog in to reply

– Priyanshu Mishra · 2 months, 3 weeks ago

Is it Geometry revisited by MAA?Log in to reply

– Ayush Rai · 2 months, 3 weeks ago

yupLog in to reply

– Priyanshu Mishra · 2 months, 3 weeks ago

That is not so good. It's good for reading only. It has less computational problems.Log in to reply

– Ayush Rai · 2 months, 3 weeks ago

yup.i use it to learn new stuff.Log in to reply

– Priyanshu Mishra · 2 months, 3 weeks ago

Do you have any good book regarding NUMBER THEORY?Log in to reply

– Ayush Rai · 2 months, 3 weeks ago

can u give ur email IDLog in to reply

– Priyanshu Mishra · 2 months, 3 weeks ago

I gave. See above.Log in to reply

– Ayush Rai · 2 months, 3 weeks ago

David M.burton book;sLog in to reply

This is there in IF Sharygin, plane geometry. – Priyanshu Mishra · 3 months, 1 week ago

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A problem copied from IF Sharygin Plane geometry. This is bad. – Priyanshu Mishra · 3 months, 1 week ago

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– Sayantan Saha · 3 months, 1 week ago

Why will I copy? This is RMO Delhi's problemLog in to reply

I am blaming RMO people that they have took this problem from a popular book which usually does not happen. – Priyanshu Mishra · 3 months, 1 week ago

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This is disappointing :( – Harsh Shrivastava · 3 months, 1 week ago

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– Priyanshu Mishra · 3 months, 1 week ago

Actually this year Delhi paper was prepared by Delhi coordinator only not by HBCSE. So he copied many things because framing a question requires a lot of time.Log in to reply

rmo dehli right – Abhishek Alva · 3 months, 1 week ago

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– Sayantan Saha · 3 months, 1 week ago

yeah.Are you appearing for this year's rmo?Log in to reply