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.:)

## 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! – Parth Lohomi · 2 years ago

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– Rajdeep Dhingra · 2 years ago

Nice question. But I already mentioned no Calculus.Log in to reply

– Parth Lohomi · 2 years ago

Oh my bad.Log in to reply

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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– Trevor Arashiro · 2 years ago

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

– Nihar Mahajan · 2 years ago

yeah... sorry.(typo)Log in to reply

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 agoLog in to reply

@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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– Rajdeep Dhingra · 2 years ago

Nice question. But this is a proving contestLog in to reply

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

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– Tanishq Varshney · 2 years ago

Ok should i remove it and type in new commentLog in to reply

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

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– Rajdeep Dhingra · 2 years ago

Maybe. But I will see to it.Log in to reply

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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– Arian Tashakkor · 2 years ago

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

– Harsh Shrivastava · 2 years ago

Level 1 or 2 maybe. What about you?What lvl will you classify this?Log in to reply

– Arian Tashakkor · 2 years ago

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)Log in to reply

– Harsh Shrivastava · 2 years ago

Lollololl :PLog in to reply

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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– Rajdeep Dhingra · 2 years ago

Nice question ! Thanks.Log in to reply

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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– Sudeep Salgia · 2 years ago

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

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

Respected sirs,

Please give questions. – Rajdeep Dhingra · 2 years ago

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