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

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

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## Comments

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TopNewestMan the deadline's over. I expected the results to come today.

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

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Ah I understand we'll wait.

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

Guys and Sirs Here is the OPC 4 !

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

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Point 1 - I am not sir. (Just 14years old).

And for the or question U have to choose one of them. $\ddot\smile$

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Just curious : - Which one have U done ?

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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?

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Go 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. $\ddot\smile$

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in no.7, it should be mentioned that the first term is not zero.

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^True dat.

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@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.

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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.

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Thanks for spotting it out. I will surely change it.

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I'm still waiting. :/

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

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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

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

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@Rajdeep Dhingra Dude.

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I have a problem with Q(5). $(b+c)/a+(c+a)/b+(a+b)/c$ should be in HP.

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Wait I will look into it. I think its AP

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@Rajdeep Dhingra I too think the question should refer to H.P. I will send you a proof for them being in H.P.

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$\ddot\smile$

I check the inbox every afternoon so U will get the reply as soon as I check it.Log in to reply