Hello Every\(\displaystyle \int_0^1{dx}\).

This is the third part of the contest started by me.Here is the \(3^{rd}\) part.The rules are the same as the previous ones. You need need to mail the PDFs to opencontestproofs@gmail.com. To make a PDF you can go to Latex PDF. Also you can send a snip of the solution after typing it in \(\LaTeX\).

Here are the questions

1.Look at the figure below :

It is given that \(AB || GF \) and \(EFGH\) is parallelogram. Prove that \(FE || DC\) .

\(2\). Look at the figure below :

\(ABCD\) is a parallelogram and \(AXY\) is a straight line through \(A\) meeting \(BC\) at \(X\) and \(DC\) at \(Y\). Prove that \(BX\times DY\) is a constant.

\(3\).Look at the figure below:

Given four points \(A,B,C,D\) in a straight line, show that a point \(O\) exists in the same straight line such that \(OA:OB = OC:OD\).Also find that point.

\(4\).Look at the figure below:

\(PQ\) is a chord of a circle. Through the mid point \(M\) of \(PQ\) chords \(AB\) and \(CD\) are drawn. \(AD\) and \(BC\) meet \(PQ\) at \(K\) and \(L\). Then prove that \(M\) is the midpoint of \(KL\).

## \[\huge \textbf{Cancelled}\]

Theme : Geometry Triangle

## Comments

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TopNewestI will participate \(\ddot\smile\) \(\ddot\smile\) \(\ddot\smile\) \(\ddot\smile\) \(\ddot\smile\) \(\ddot\smile\) – Mehul Arora · 2 years, 2 months ago

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– Azhaghu Roopesh M · 2 years, 1 month ago

Yeah ,it's correct . Looks like you'll win this contest for sure :PLog in to reply

– Nihar Mahajan · 2 years, 1 month ago

Well , Euclid will become angry and will come for revenge on seeing my uploaded diagram above. :P :D xDLog in to reply

– Azhaghu Roopesh M · 2 years, 1 month ago

No he won't . There's only one \(\color{red}\text{Euclidean}\) error in it . Just add two arrow heads on the ends of the line and you're done :)Log in to reply

– Omkar Kulkarni · 2 years, 2 months ago

Even I thought there was something wrong with the problem! Your proof is flawless. Just to clarify, ahead of \(E\) being the midpoint of \(AC\), you can say that \(E\) was chosen to lie on \(XC\). Hence, \(A\) and \(B\) are coincident and the triangle cannot be formed.Log in to reply

– Rajdeep Dhingra · 2 years, 2 months ago

Q4 has been updated.Log in to reply

– Nihar Mahajan · 2 years, 1 month ago

I have mad some minor mistakes while giving solutions.Please make sure you read the corrections that I have sent.Log in to reply

– Omkar Kulkarni · 2 years, 2 months ago

Yup thanks! It seems pretty interesting :)Log in to reply

– Rajdeep Dhingra · 2 years, 2 months ago

Q4 has been updated.Log in to reply

– Nihar Mahajan · 2 years, 2 months ago

Thanks a lot!Log in to reply

– Rajdeep Dhingra · 2 years, 2 months ago

Welcome.Log in to reply

– Nihar Mahajan · 2 years, 2 months ago

I think I have seen this somewhere... Did you create these problems?Log in to reply

– Rajdeep Dhingra · 2 years, 2 months ago

This one not. It is a very famous theorem.Log in to reply

– Nihar Mahajan · 2 years, 2 months ago

I don't know its name , but I had proved this in 8th Std. But I don't remember the proof now , So, i have to rethink now :)Log in to reply

– Rajdeep Dhingra · 2 years, 2 months ago

Please delete the comment with contradiction to Q4Log in to reply

– Nihar Mahajan · 2 years, 2 months ago

I think , it must be deleted after 24 hours , or the people who are solving or have taken note of the previous problem will start questioning 'why it is removed , where it has gone?' .Hence I prefer deleting it after 24 hours.Log in to reply

– Rajdeep Dhingra · 2 years, 2 months ago

Yes you are right. I am thinking of removing it. I will add another one in place of it.Log in to reply

@Omkar Kulkarni 's median contradiction. – Nihar Mahajan · 2 years, 2 months ago

Actually this suspicion came byLog in to reply

– Rajdeep Dhingra · 2 years, 1 month ago

Is Q4 fine ?Log in to reply

– Nihar Mahajan · 2 years, 1 month ago

Yeah its good , I got it!Log in to reply

– Rajdeep Dhingra · 2 years, 1 month ago

Are you going to participate ? Deadlines nearLog in to reply

– Omkar Kulkarni · 2 years, 1 month ago

I give up on the third one, I'll try to send the others before 10:29! I completely forgot about coding the solutions. :PLog in to reply

– Nihar Mahajan · 2 years, 1 month ago

Yeah , There was some problem in my latex , so I cannot send the solution for 3rd problem.No option :/ I will send solutions to only problem number 1,2,4.Log in to reply

@Rajdeep Dhingra , When Will you give out the results? -_- – Mehul Arora · 2 years, 1 month ago

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– Nihar Mahajan · 2 years, 1 month ago

This contest is cancelled and we all are trolled ... xD Cheers! :( :(Log in to reply

– Omkar Kulkarni · 2 years, 1 month ago

Could you post a solution for the third one? It is troubling me SOO much.Log in to reply

– Mehul Arora · 2 years, 1 month ago

Why the hell is the Contest Cancelled? o.OLog in to reply

– Rajdeep Dhingra · 2 years, 1 month ago

Sorry for that. But since Schools are going on and I am very much busy, I can't Compile the result. I will release OPC 4 on 18th May. Do participate in that one. Sorry.Log in to reply

– Mehul Arora · 2 years, 1 month ago

Okay. No problem. :)Log in to reply

Rajdeep would you suggest some books for RMO – Samruddh Kamath · 2 years, 1 month ago

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@Rajdeep Dhingra I had error with converting Latex to pdf.So i managed to convert Latex to photo. Is it ok if I send you photos? – Nihar Mahajan · 2 years, 1 month ago

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– Rajdeep Dhingra · 2 years, 1 month ago

Yes. You can send the photos.Log in to reply

@Rajdeep Dhingra , Can I Give my solution in MS Word? I am not really familiar with LaTeX Yet. Also, Can I give my solution in Parts? I have a lot of Homework To do. – Mehul Arora · 2 years, 1 month ago

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– Rajdeep Dhingra · 2 years, 1 month ago

Yes, Of course.Log in to reply

– Mehul Arora · 2 years, 1 month ago

Okay :)Log in to reply

I will participate – Samruddh Kamath · 2 years, 1 month ago

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– Rajdeep Dhingra · 2 years, 1 month ago

Deadline is NearLog in to reply

– Rajdeep Dhingra · 2 years, 1 month ago

All the best !Log in to reply

Could you provide a link to the previous ones, so as to give everyone easy access to the rules? Thanks. I'll participate too :) – Omkar Kulkarni · 2 years, 2 months ago

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– Rajdeep Dhingra · 2 years, 1 month ago

Deadline is nearLog in to reply

OPC2 – Rajdeep Dhingra · 2 years, 2 months ago

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@Rajdeep Dhingra

Where is

Gin the figure to the first question ? – Azhaghu Roopesh M · 2 years, 2 months agoLog in to reply

– Rajdeep Dhingra · 2 years, 2 months ago

G is the point on the parallelogram. GH is parallel to EFLog in to reply

– Azhaghu Roopesh M · 2 years, 2 months ago

I understood that , but won't you make an edit in the picture ?Log in to reply

– Rajdeep Dhingra · 2 years, 2 months ago

Right now I have to go somewhere. So I will make it in about an hour.Log in to reply

– Azhaghu Roopesh M · 2 years, 2 months ago

Yeah , fine .Log in to reply

– Rajdeep Dhingra · 2 years, 2 months ago

Done it.Log in to reply

dotthat's come out of nowhere ;) – Azhaghu Roopesh M · 2 years, 2 months agoLog in to reply

– Rajdeep Dhingra · 2 years, 2 months ago

Yes. Just a editing mistake.Log in to reply

– Azhaghu Roopesh M · 2 years, 2 months ago

Ok .Log in to reply

@Sandeep Bhardwaj @Azhaghu Roopesh M @megh choksi @Daniel Liu @Prasun Biswas @Parth Bhardwaj @Cody Martin – Rajdeep Dhingra · 2 years, 2 months ago

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@Rajdeep Dhingra – Sandeep Bhardwaj · 2 years, 2 months ago

All the best dear. You're making it Great. Keep it up.Log in to reply

– Rajdeep Dhingra · 2 years, 2 months ago

Thank you sir ! You always motivate people. Proud to be your student.Log in to reply

– Azhaghu Roopesh M · 2 years, 2 months ago

You noticed it too ! That Megh was online today .Log in to reply

– Rajdeep Dhingra · 2 years, 2 months ago

YesLog in to reply

@Azhaghu Roopesh M Let's see. :) :) :) – Nihar Mahajan · 2 years, 2 months ago

I am also participating.Log in to reply

– Azhaghu Roopesh M · 2 years, 2 months ago

Challenge accepted :PLog in to reply

– Azhaghu Roopesh M · 2 years, 2 months ago

Yes . Though I'll submit my entries after the 4th .Log in to reply

– Rajdeep Dhingra · 2 years, 2 months ago

No Problem. Best of luck.Log in to reply

– Azhaghu Roopesh M · 2 years, 2 months ago

Thanks :)Log in to reply