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Open Proof Contest 3

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

Note by Rajdeep Dhingra
1 year, 12 months ago

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I will participate \(\ddot\smile\) \(\ddot\smile\) \(\ddot\smile\) \(\ddot\smile\) \(\ddot\smile\) \(\ddot\smile\) Mehul Arora · 1 year, 12 months ago

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Comment deleted Apr 12, 2015

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@Nihar Mahajan Yeah ,it's correct . Looks like you'll win this contest for sure :P Azhaghu Roopesh M · 1 year, 12 months ago

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@Azhaghu Roopesh M Well , Euclid will become angry and will come for revenge on seeing my uploaded diagram above. :P :D xD Nihar Mahajan · 1 year, 12 months ago

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@Nihar Mahajan 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 :) Azhaghu Roopesh M · 1 year, 12 months ago

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@Nihar Mahajan 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. Omkar Kulkarni · 1 year, 12 months ago

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@Omkar Kulkarni Q4 has been updated. Rajdeep Dhingra · 1 year, 12 months ago

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@Rajdeep Dhingra I have mad some minor mistakes while giving solutions.Please make sure you read the corrections that I have sent. Nihar Mahajan · 1 year, 11 months ago

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@Rajdeep Dhingra Yup thanks! It seems pretty interesting :) Omkar Kulkarni · 1 year, 12 months ago

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@Nihar Mahajan Q4 has been updated. Rajdeep Dhingra · 1 year, 12 months ago

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@Rajdeep Dhingra Thanks a lot! Nihar Mahajan · 1 year, 12 months ago

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@Nihar Mahajan Welcome. Rajdeep Dhingra · 1 year, 12 months ago

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@Rajdeep Dhingra I think I have seen this somewhere... Did you create these problems? Nihar Mahajan · 1 year, 12 months ago

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@Nihar Mahajan This one not. It is a very famous theorem. Rajdeep Dhingra · 1 year, 12 months ago

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@Rajdeep Dhingra 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 :) Nihar Mahajan · 1 year, 12 months ago

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@Nihar Mahajan Please delete the comment with contradiction to Q4 Rajdeep Dhingra · 1 year, 12 months ago

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@Rajdeep Dhingra 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. Nihar Mahajan · 1 year, 12 months ago

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@Nihar Mahajan Yes you are right. I am thinking of removing it. I will add another one in place of it. Rajdeep Dhingra · 1 year, 12 months ago

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@Rajdeep Dhingra Actually this suspicion came by @Omkar Kulkarni 's median contradiction. Nihar Mahajan · 1 year, 12 months ago

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@Nihar Mahajan Is Q4 fine ? Rajdeep Dhingra · 1 year, 11 months ago

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@Rajdeep Dhingra Yeah its good , I got it! Nihar Mahajan · 1 year, 11 months ago

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@Nihar Mahajan Are you going to participate ? Deadlines near Rajdeep Dhingra · 1 year, 11 months ago

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@Rajdeep Dhingra I give up on the third one, I'll try to send the others before 10:29! I completely forgot about coding the solutions. :P Omkar Kulkarni · 1 year, 11 months ago

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@Rajdeep Dhingra 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. Nihar Mahajan · 1 year, 11 months ago

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@Rajdeep Dhingra , When Will you give out the results? -_- Mehul Arora · 1 year, 11 months ago

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@Mehul Arora This contest is cancelled and we all are trolled ... xD Cheers! :( :( Nihar Mahajan · 1 year, 11 months ago

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@Nihar Mahajan Could you post a solution for the third one? It is troubling me SOO much. Omkar Kulkarni · 1 year, 11 months ago

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@Nihar Mahajan Why the hell is the Contest Cancelled? o.O Mehul Arora · 1 year, 11 months ago

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@Mehul Arora 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. Rajdeep Dhingra · 1 year, 11 months ago

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@Rajdeep Dhingra Okay. No problem. :) Mehul Arora · 1 year, 11 months ago

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Rajdeep would you suggest some books for RMO Samruddh Kamath · 1 year, 11 months 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 · 1 year, 11 months ago

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@Nihar Mahajan Yes. You can send the photos. Rajdeep Dhingra · 1 year, 11 months ago

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@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 · 1 year, 11 months ago

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@Mehul Arora Yes, Of course. Rajdeep Dhingra · 1 year, 11 months ago

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@Rajdeep Dhingra Okay :) Mehul Arora · 1 year, 11 months ago

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I will participate Samruddh Kamath · 1 year, 11 months ago

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@Samruddh Kamath Deadline is Near Rajdeep Dhingra · 1 year, 11 months ago

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@Samruddh Kamath All the best ! Rajdeep Dhingra · 1 year, 11 months ago

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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 · 1 year, 12 months ago

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@Omkar Kulkarni Deadline is near Rajdeep Dhingra · 1 year, 11 months ago

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@Omkar Kulkarni OPC2 Rajdeep Dhingra · 1 year, 12 months ago

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@Rajdeep Dhingra

Where is G in the figure to the first question ? Azhaghu Roopesh M · 1 year, 12 months ago

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@Azhaghu Roopesh M G is the point on the parallelogram. GH is parallel to EF Rajdeep Dhingra · 1 year, 12 months ago

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@Rajdeep Dhingra I understood that , but won't you make an edit in the picture ? Azhaghu Roopesh M · 1 year, 12 months ago

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@Azhaghu Roopesh M Right now I have to go somewhere. So I will make it in about an hour. Rajdeep Dhingra · 1 year, 12 months ago

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@Rajdeep Dhingra Yeah , fine . Azhaghu Roopesh M · 1 year, 12 months ago

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@Azhaghu Roopesh M Done it. Rajdeep Dhingra · 1 year, 12 months ago

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@Rajdeep Dhingra Thanks :) But it seems there's an extra dot that's come out of nowhere ;) Azhaghu Roopesh M · 1 year, 12 months ago

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@Azhaghu Roopesh M Yes. Just a editing mistake. Rajdeep Dhingra · 1 year, 12 months ago

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@Rajdeep Dhingra Ok . Azhaghu Roopesh M · 1 year, 12 months ago

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@Rajdeep Dhingra All the best dear. You're making it Great. Keep it up. @Rajdeep Dhingra Sandeep Bhardwaj · 1 year, 12 months ago

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@Sandeep Bhardwaj Thank you sir ! You always motivate people. Proud to be your student. Rajdeep Dhingra · 1 year, 12 months ago

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@Rajdeep Dhingra You noticed it too ! That Megh was online today . Azhaghu Roopesh M · 1 year, 12 months ago

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@Azhaghu Roopesh M Yes Rajdeep Dhingra · 1 year, 12 months ago

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@Rajdeep Dhingra I am also participating. @Azhaghu Roopesh M Let's see. :) :) :) Nihar Mahajan · 1 year, 12 months ago

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@Nihar Mahajan Challenge accepted :P Azhaghu Roopesh M · 1 year, 12 months ago

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@Rajdeep Dhingra Yes . Though I'll submit my entries after the 4th . Azhaghu Roopesh M · 1 year, 12 months ago

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@Azhaghu Roopesh M No Problem. Best of luck. Rajdeep Dhingra · 1 year, 12 months ago

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@Rajdeep Dhingra Thanks :) Azhaghu Roopesh M · 1 year, 12 months ago

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