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.:)

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## Comments

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TopNewest1) 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!

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Nice question. But I already mentioned no Calculus.

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Oh my bad.

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A intermediate problem by me ,

Q 1) Find all '\(a\)' such that \(24\ | \ a^2+34a+1919\)

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I think u mean that to be an \(a^2\)

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yeah... sorry.(typo)

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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?

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@Rajdeep Dhingra , check out this algebra question.

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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 .

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Nice question. But this is a proving contest

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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 \(a,b,c>0\)

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Note: Type new questions in new comments

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

I guess that makes it a proof :3

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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 \(n\).

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An easy proof problem :

Prove that \(n!\) > \(2^{n}\) for all integers \(n \geq 4\).

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Just out of curiosity , what level would you classify this question in?

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Level 1 or 2 maybe. What about you?What lvl will you classify this?

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A slightly-above-intermediate problem by me.

Q #1 .

\(\quad\)

Find all integral solutions to the following equation:

\(\quad\)

\(\frac {1}{m} + \frac{1}{n} - \frac {1} { mn^2} = \frac {3}{4} \)

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Nice question ! Thanks.

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Lemme post a geometry problem :

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

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@Archit Boobna @Parth Bhardwaj see this

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I think you meant 24th May in the deadline. I have corrected it accordingly.

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

Respected sirs,

Please give questions.

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