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Open Proof Contest 4 - Questions Needed

Hello Every\( \displaystyle \int_{0}^{\pi/2}{\cos{x} dx}\)

This is the 4th and the best part of the contest started by me. It's Open Proof Contest. This time the contest will contain 20 questions out of which 10 will be mine and 10 from U all. Don't worry even if U give a question and I put it in the Contest then also u can participate. Those 10 questions will have choices. See each question which will be from you will have a choice abc or xyz. If your question is abc then U have to do xyz and vice versa.

There is no particular topic.U can give any questions except calculus. Give your questions in seperate comments over here. Any doubts feel free to ask. Also don't give the solution to Ur question. until U are asked to. Open till 23th May 11: 59 IST.

Do Reshare

Hoping to see many great questions.:)

Note by Rajdeep Dhingra
2 years ago

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1) Prove that

\[\displaystyle\int_0^{\infty} \ dx \ln^2 \tanh (x) =\dfrac {7}{4}\zeta {(3) }\]

\(\tanh(x) \) is the hypebolic trigonometry function


You will be provided the solution if you want it. Cheers! Parth Lohomi · 2 years ago

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@Parth Lohomi Nice question. But I already mentioned no Calculus. Rajdeep Dhingra · 2 years ago

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@Rajdeep Dhingra Oh my bad. Parth Lohomi · 2 years ago

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A intermediate problem by me ,

Q 1) Find all '\(a\)' such that \(24\ | \ a^2+34a+1919\) Nihar Mahajan · 2 years ago

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@Nihar Mahajan I think u mean that to be an \(a^2\) Trevor Arashiro · 2 years ago

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@Trevor Arashiro yeah... sorry.(typo) Nihar Mahajan · 2 years ago

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You want algebra problem?

Let \(a,b,c\) be positive reals satisfying \(a^3+b^3+c^3+abc=4\) . Prove that \[ \frac{(5a^2+bc)^2}{(a+b)(a+c)} + \frac{(5b^2+ca)^2}{(b+c)(b+a)} + \frac{(5c^2+ab)^2}{(c+a)(c+b)} \ge \frac{(a^3+b^3+c^3+6)^2}{a+b+c} \] .

BONUS : How many cases of equality are there?
Parth Lohomi · 2 years ago

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@Parth Lohomi @Rajdeep Dhingra , check out this algebra question. Parth Lohomi · 2 years ago

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A intermediate problem by me,

Q1) On a chess-board if two squares are chosen, what is the probability that they have side in common . Tanishq Varshney · 2 years ago

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@Tanishq Varshney Nice question. But this is a proving contest Rajdeep Dhingra · 2 years ago

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@Rajdeep Dhingra Ok here is my question

prove that \(\large{\frac{a^{8}+b^{8}+c^{8}}{a^{3}b^{3}c^{3}}>\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}\) where \(a,b,c>0\) Tanishq Varshney · 2 years ago

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@Tanishq Varshney Great! This one surely will make it.

Note: Type new questions in new comments Rajdeep Dhingra · 2 years ago

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@Rajdeep Dhingra Ok should i remove it and type in new comment Tanishq Varshney · 2 years ago

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@Rajdeep Dhingra Just reword it to "prove that the probability is __".

I guess that makes it a proof :3 Trevor Arashiro · 2 years ago

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@Trevor Arashiro Maybe. But I will see to it. Rajdeep Dhingra · 2 years ago

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This is a nice question but not original :

Q #2) \(n\) circles are drawn with a radii length of 1 unit where \( n \ge 2\) , such that every two circles intersect.Prove that the number of intersection points is at least \(n\). Arian Tashakkor · 2 years ago

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An easy proof problem :

Prove that \(n!\) > \(2^{n}\) for all integers \(n \geq 4\). Harsh Shrivastava · 2 years ago

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@Harsh Shrivastava Just out of curiosity , what level would you classify this question in? Arian Tashakkor · 2 years ago

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@Arian Tashakkor Level 1 or 2 maybe. What about you?What lvl will you classify this? Harsh Shrivastava · 2 years ago

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@Harsh Shrivastava Level 2-3.IMO , the method of solving this problem is not so trivial to be classified as a level 1 or 2.(I'm trying not to give away the solving method I think one may "INDUCT" electricity otherwise?! xD) Arian Tashakkor · 2 years ago

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@Arian Tashakkor Lollololl :P Harsh Shrivastava · 2 years ago

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A slightly-above-intermediate problem by me.

Q #1 .

\(\quad\)

Find all integral solutions to the following equation:

\(\quad\)

\(\frac {1}{m} + \frac{1}{n} - \frac {1} { mn^2} = \frac {3}{4} \) Arian Tashakkor · 2 years ago

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@Arian Tashakkor Nice question ! Thanks. Rajdeep Dhingra · 2 years ago

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Lemme post a geometry problem :

In \(\Delta ABC\) , if \(BC^2=AC(AC+AB)\) , then prove that \(m\angle CAB = 2 \times m\angle CBA\). Nihar Mahajan · 2 years ago

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@Archit Boobna @Parth Bhardwaj see this Rajdeep Dhingra · 2 years ago

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@Rajdeep Dhingra I think you meant 24th May in the deadline. I have corrected it accordingly. Sudeep Salgia · 2 years ago

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@Sandeep Bhardwaj @Calvin Lin @Brian Charlesworth @Pi Han Goh

Respected sirs,
Please give questions. Rajdeep Dhingra · 2 years ago

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