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Open Proof Contest 4 - Summer Time is Proving time

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

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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 :

  1. Find all 'a' such that 24 divides \(a^2 +34a + 1919\) \(\rightarrow \text{By NIHAR}\) \[OR\]
    Find all integral solutions to the following equation: \( \frac{1}{m} + \frac{1}{n} - \frac{1}{mn^2} = \frac{3}{4}\) \(\rightarrow \text{By ARIAN} \).

  2. 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

  3. \(n\) 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

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

  5. 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..

  6. 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 \(\frac{1}{n}\).

  7. 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 \(\dfrac{2m-1}{2n-1}\).

  8. 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.

  9. 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'.

  10. 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

Note by Rajdeep Dhingra
2 years, 4 months ago

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Man the deadline's over. I expected the results to come today. Kunal Verma · 2 years, 3 months ago

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@Kunal Verma 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 Rajdeep Dhingra · 2 years, 3 months ago

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@Rajdeep Dhingra Ah I understand we'll wait. Kunal Verma · 2 years, 3 months ago

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@Rajdeep Dhingra Dude. Kunal Verma · 2 years, 3 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 · 2 years, 3 months ago

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@Manish Dash Just send me a mail (On my real ID(the "rajdeep" one)) and I will see where the problem is. Rajdeep Dhingra · 2 years, 3 months ago

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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. Kunal Verma · 2 years, 3 months ago

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@Kunal Verma Thanks for spotting it out. I will surely change it. Rajdeep Dhingra · 2 years, 3 months ago

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@Rajdeep Dhingra Dude you haven't opened the proof contest ID in a while. I thought you opened it every afternoon. Kunal Verma · 2 years, 3 months ago

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@Kunal Verma Busy, will open when free. Rajdeep Dhingra · 2 years, 3 months ago

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@Rajdeep Dhingra I'm still waiting. :/ Kunal Verma · 2 years, 3 months ago

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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. Kunal Verma · 2 years, 3 months ago

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in no.7, it should be mentioned that the first term is not zero. Aareyan Manzoor · 2 years, 3 months ago

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@Aareyan Manzoor ^True dat. Kunal Verma · 2 years, 3 months ago

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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? Kunal Verma · 2 years, 3 months ago

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@Kunal Verma 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. \(\ddot\smile\) Rajdeep Dhingra · 2 years, 3 months 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, 4 months ago

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@Rajdeep Dhingra Sir, for the OR question, do we choose only one or we do both of them? Figel Ilham · 2 years, 4 months ago

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@Figel Ilham Point 1 - I am not sir. (Just 14years old).
And for the or question U have to choose one of them. \(\ddot\smile\) Rajdeep Dhingra · 2 years, 4 months ago

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@Rajdeep Dhingra For writing in latex, what font, what size and what paper for OPC-4? Figel Ilham · 2 years, 4 months ago

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@Figel Ilham Any. Just the sol should be clear. It can be even in Ur handwriting. Rajdeep Dhingra · 2 years, 4 months ago

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@Rajdeep Dhingra 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? Figel Ilham · 2 years, 4 months ago

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@Figel Ilham Thanks. Keep on trying they are very easy. Especially 8,4,51,2(or).
Just curious : - Which one have U done ? Rajdeep Dhingra · 2 years, 4 months ago

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@Rajdeep Dhingra Ineq 2 (Tanishq Varshney) Really easy? I don't get any solutions for number 1 or it is said null. Figel Ilham · 2 years, 4 months ago

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I have a problem with Q(5). \((b+c)/a+(c+a)/b+(a+b)/c\) should be in HP. Satvik Pandey · 2 years, 4 months ago

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@Satvik Pandey Wait I will look into it. I think its AP Rajdeep Dhingra · 2 years, 4 months ago

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@Rajdeep Dhingra @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, 3 months ago

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@Kunal Verma Thanks I will change it. Rajdeep Dhingra · 2 years, 3 months ago

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@Rajdeep Dhingra Btw what about the other solutions I sent. I did not get any replies regarding them. Kunal Verma · 2 years, 3 months ago

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@Kunal Verma I check the inbox every afternoon so U will get the reply as soon as I check it. \(\ddot\smile\) Rajdeep Dhingra · 2 years, 3 months ago

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Comment deleted May 27, 2015

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@Swapnil Das And you are getting really jealous. @Swapnil Das Nitesh Chaudhary · 2 years, 3 months ago

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@Nitesh Chaudhary Can U tell me what he said ? Just curious Rajdeep Dhingra · 2 years, 3 months ago

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@Rajdeep Dhingra And they say I am getting jealous! Swapnil Das · 2 years, 3 months ago

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@Rajdeep Dhingra I just said : "Hey Rajdeep, U are getting quiet famous, so do not forget us!" Swapnil Das · 2 years, 3 months ago

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@Rajdeep Dhingra By the statement that Swapnil Das wrote , anybody can think that he is jealous of you. Nitesh Chaudhary · 2 years, 3 months ago

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