\[ \large \color{red}{OPEN}\quad \color{blue}{PROOF} \quad \color{green}{CONTEST} \color{red}{- 4} \]

Hello Every\(\quad \displaystyle \dfrac{4}{7 \zeta(3)}\int_{0}^{\infty}{\ln^2{ \tanh(x)} dx}\)

This is the 4\(^{th}\) Part of the contest started by me. It is the Open Proof Contest! This time there will be 10 questions(3 from U and 7 are mine ). The rules remain the same except for that the persons who's questions are listed have to do the 'or' one part. You have to mail the PDFs or images or documents to opencontestproofs@gmail.com. You can submit the solution in parts. To make a PDF you can go to TeXeR.

Here are the questions :

Find all 'a' such that 24 divides \(a^2 +34a + 1919\) \(\rightarrow \text{By NIHAR}\) \[OR\]

Find all integral solutions to the following equation: \( \frac{1}{m} + \frac{1}{n} - \frac{1}{mn^2} = \frac{3}{4}\) \(\rightarrow \text{By ARIAN} \).Let \(a,b,c\) be positive reals satisfying \(a^3 + b^3 + c^3 + abc = 4\). Prove that \[\frac{(5a^2+bc)^2}{(a+b)(a+c)} + \frac{(5b^2+ca)^2}{(b+c)(b+a)} + \frac{(5c^2+ab)^2}{(c+a)(c+b)} \ge \frac{(a^3+b^3+c^3+6)^2}{a+b+c}\] By Parth Lohmi \[OR\] 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 \(a,b,c>0\) by Tanishq Varshney

\(n\) 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 \(n\). By Arian \[OR\] In \(\Delta ABC\) , if \(BC^2=AC(AC+AB)\) , then prove that \(m\angle CAB = 2 \times m\angle CBA\). By Nihar

Find all polynomials such that \(p(x+2) = p(x) + 4x + 4\).

If \(a^2 , b^2 , c^2\) are in AP then prove that \(\frac{b+c}{a},\frac{c+a}{b},\frac{a+b}{c}\) are in HP..

Let x be any real number. Prove that among the numbers \(x,2x,....,(n-1)x\) there is one number that differs from an integer by at most \(\frac{1}{n}\).

The sum of r terms of an AP is denoted by \(S_{r}\). And \(\dfrac{S_{m}}{S_{n}} = \frac{m^2}{n^2}\). Prove that the ratio of m\(^{th}\) term to n\(^{th}\) term is \(\dfrac{2m-1}{2n-1}\).

Consider all lines which meet the graph of \(y = 10! x^{10} - 9! x^9 + 8! x^8 - 7! x^7 + ..... + 2! x^2 - 1! x + 0!\) in 10 distinct points say \((x_1,y_1), ..... ,(x_{10},y_{10})\). Then show that the value of \(\displaystyle \dfrac{\displaystyle \sum_{i = 1}^{10}{x_i}}{10}\) is independent of the line and also give its value.

Let p > 3 be an odd prime. Suppose \(\displaystyle \sum_{k = 1}^{p-1}{\dfrac{1}{k}} = \frac{a}{b}\), where GCD(a,b) = 1. Prove that 'a' is divisble by 'p'.

If a,b are +ve integers such that the number \(\dfrac{a+1}{b} + \dfrac{b+1}{a}\) is also an integer, then prove that GCD(a,b)\(\leq \sqrt{a+b} \).

Do Reshare and Participate. This is the best one till now and probably will be help next year after this.

Deadline 10th June 2015

U can also go here

## Comments

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TopNewestMan the deadline's over. I expected the results to come today. – Kunal Verma · 1 year, 11 months ago

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– Rajdeep Dhingra · 1 year, 11 months ago

Sorry for the inconvinience , I have a phase test this week and there are 15 entries to see. I will surely do it by mondayLog in to reply

– Kunal Verma · 1 year, 11 months ago

Ah I understand we'll wait.Log in to reply

@Rajdeep Dhingra Dude. – Kunal Verma · 1 year, 11 months ago

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Well, how do I end the document? When I click on render as PDF, I get the message "Error compiling LaTeX. ! LaTeX Error: There's no line here to end". Please help me @Rajdeep Dhingra and others – Manish Dash · 1 year, 12 months ago

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– Rajdeep Dhingra · 1 year, 12 months ago

Just send me a mail (On my real ID(the "rajdeep" one)) and I will see where the problem is.Log in to reply

I have two doubts.

1) Q2) in Tanishq's alternative, if a = b= c, then equality between the given expressions holds.

2) Q3) in Arian's alternative, does intersecting of circles at one single point i.e. the circles touching each other count? Because if it does, then there will always be a case where there are \(n \ - \ 1\) intersection points for n drawn circles. – Kunal Verma · 1 year, 12 months ago

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– Rajdeep Dhingra · 1 year, 12 months ago

Thanks for spotting it out. I will surely change it.Log in to reply

– Kunal Verma · 1 year, 12 months ago

Dude you haven't opened the proof contest ID in a while. I thought you opened it every afternoon.Log in to reply

– Rajdeep Dhingra · 1 year, 12 months ago

Busy, will open when free.Log in to reply

– Kunal Verma · 1 year, 12 months ago

I'm still waiting. :/Log in to reply

@Rajdeep Dhingra You didn't even check your mail today. :/ I managed to pull off 6 and 9 and I really wanted you to look at them. – Kunal Verma · 2 years ago

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in no.7, it should be mentioned that the first term is not zero. – Aareyan Manzoor · 2 years ago

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– Kunal Verma · 2 years ago

^True dat.Log in to reply

I'm having problem in ending the latex on AoPS. It's getting annoying everytime I see "There's no line to end." when I click Render as PDF. Is there any other way? – Kunal Verma · 2 years ago

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here and paste the LateX. Take a screen shot or click save to Google Drive and mail me the Image. Or U could Just check whether Ur \(\LaTeX\) is right or not and try again. \(\ddot\smile\) – Rajdeep Dhingra · 2 years ago

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@Archit Boobna @Parth Bhardwaj @Sandeep Bhardwaj @Calvin Lin

Guys and Sirs Here is the OPC 4 ! – Rajdeep Dhingra · 2 years ago

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– Figel Ilham · 2 years ago

Sir, for the OR question, do we choose only one or we do both of them?Log in to reply

And for the or question U have to choose one of them. \(\ddot\smile\) – Rajdeep Dhingra · 2 years ago

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– Figel Ilham · 2 years ago

For writing in latex, what font, what size and what paper for OPC-4?Log in to reply

– Rajdeep Dhingra · 2 years ago

Any. Just the sol should be clear. It can be even in Ur handwriting.Log in to reply

– Figel Ilham · 2 years ago

U are really genius and mastermind. The problems are insane. Currently, I only did one. For writing in latex, what font, what size and what paper for OPC-4?Log in to reply

Just curious : - Which one have U done ? – Rajdeep Dhingra · 2 years ago

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– Figel Ilham · 2 years ago

Ineq 2 (Tanishq Varshney) Really easy? I don't get any solutions for number 1 or it is said null.Log in to reply

I have a problem with Q(5). \((b+c)/a+(c+a)/b+(a+b)/c\) should be in HP. – Satvik Pandey · 2 years ago

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– Rajdeep Dhingra · 2 years ago

Wait I will look into it. I think its APLog in to reply

@Rajdeep Dhingra I too think the question should refer to H.P. I will send you a proof for them being in H.P. – Kunal Verma · 2 years ago

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– Rajdeep Dhingra · 2 years ago

Thanks I will change it.Log in to reply

– Kunal Verma · 2 years ago

Btw what about the other solutions I sent. I did not get any replies regarding them.Log in to reply

– Rajdeep Dhingra · 2 years ago

I check the inbox every afternoon so U will get the reply as soon as I check it. \(\ddot\smile\)Log in to reply

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@Swapnil Das – Nitesh Chaudhary · 2 years ago

And you are getting really jealous.Log in to reply

– Rajdeep Dhingra · 2 years ago

Can U tell me what he said ? Just curiousLog in to reply

– Swapnil Das · 2 years ago

And they say I am getting jealous!Log in to reply

– Swapnil Das · 2 years ago

I just said : "Hey Rajdeep, U are getting quiet famous, so do not forget us!"Log in to reply

– Nitesh Chaudhary · 2 years ago

By the statement that Swapnil Das wrote , anybody can think that he is jealous of you.Log in to reply