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Circles

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.

Note by Sayantan Saha
1 year, 2 months ago

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Problem 1

Problem 1

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 - 1 year, 1 month ago

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Very good solution. Very much clear also. I understood it now.

Priyanshu Mishra - 1 year, 1 month ago

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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??

Ayush Rai - 1 year, 1 month ago

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@Ayush Rai I have many interesting books regarding IMO. I post problems from there only.

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 - 1 year, 1 month ago

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@Priyanshu Mishra Please share these resources with me also.

My email id : harshus99@gmail.com

Harsh Shrivastava - 1 year, 1 month ago

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@Priyanshu Mishra and also give the INMO paper to this email.Did u write RMO this year

Ayush Rai - 1 year, 1 month ago

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@Ayush Rai Yes I appeared in RMO this year from KVS region.

Priyanshu Mishra - 1 year, 1 month ago

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@Priyanshu Mishra How many did u solve?

Ayush Rai - 1 year, 1 month ago

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@Ayush Rai I solved 4 and half.

Priyanshu Mishra - 1 year, 1 month ago

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@Priyanshu Mishra Awesome! Send me the question paper also then

Ayush Rai - 1 year, 1 month ago

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@Ayush Rai You will get it.

I am currently out of Delhi. I will send you after 5 Nov.

Priyanshu Mishra - 1 year, 1 month ago

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@Priyanshu Mishra ok.no probs. Can u give ur email address

Ayush Rai - 1 year, 1 month ago

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@Ayush Rai priyanshu_2feb@rediffmail.com

Priyanshu Mishra - 1 year, 1 month ago

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@Priyanshu Mishra Sometimes co-exter or pre-college maths.Can u give the pdf of the books to this email : raiayush234@gmail.com

Ayush Rai - 1 year, 1 month ago

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@Ayush Rai and @Harsh Shrivastava

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

Check your mail.

Priyanshu Mishra - 1 year, 1 month ago

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Thanks a lot bro.

Ayush Rai - 1 year, 1 month ago

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@Harsh Shrivastava, @Ayush Rai,

I have 36 Olympiad books related to IMO. Do you want all of them?

Priyanshu Mishra - 1 year, 1 month ago

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Yeah I also want em all.

Harsh Shrivastava - 1 year, 1 month ago

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yup.sure.send it to my email.

Ayush Rai - 1 year, 1 month ago

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Ok no problem.

Can you tell me the author of co-exter book?

Priyanshu Mishra - 1 year, 1 month ago

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@Priyanshu Mishra H. S. M. Coxeter and S. L. Greitzer

Ayush Rai - 1 year, 1 month ago

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@Ayush Rai Is it Geometry revisited by MAA?

Priyanshu Mishra - 1 year, 1 month ago

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@Priyanshu Mishra yup

Ayush Rai - 1 year, 1 month ago

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@Ayush Rai That is not so good. It's good for reading only. It has less computational problems.

Priyanshu Mishra - 1 year, 1 month ago

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@Priyanshu Mishra yup.i use it to learn new stuff.

Ayush Rai - 1 year, 1 month ago

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@Ayush Rai Do you have any good book regarding NUMBER THEORY?

Priyanshu Mishra - 1 year, 1 month ago

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@Priyanshu Mishra can u give ur email ID

Ayush Rai - 1 year, 1 month ago

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@Ayush Rai I gave. See above.

Priyanshu Mishra - 1 year, 1 month ago

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@Priyanshu Mishra David M.burton book;s

Ayush Rai - 1 year, 1 month ago

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This is there in IF Sharygin, plane geometry.

Priyanshu Mishra - 1 year, 2 months ago

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A problem copied from IF Sharygin Plane geometry. This is bad.

Priyanshu Mishra - 1 year, 2 months ago

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Why will I copy? This is RMO Delhi's problem

Sayantan Saha - 1 year, 2 months ago

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What I am saying and what are you understanding.

I am blaming RMO people that they have took this problem from a popular book which usually does not happen.

Priyanshu Mishra - 1 year, 2 months ago

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@Priyanshu Mishra The inequality problem in Delhi region was also copied.I don't remember the source, but I had seen it before RMO on some forum.

This is disappointing :(

Harsh Shrivastava - 1 year, 2 months ago

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@Harsh Shrivastava 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.

Priyanshu Mishra - 1 year, 2 months ago

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rmo dehli right

Abhishek Alva - 1 year, 2 months ago

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yeah.Are you appearing for this year's rmo?

Sayantan Saha - 1 year, 2 months ago

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