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.

U can also go here

Note by Rajdeep Dhingra
6 years, 2 months ago

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

- 6 years, 1 month 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 monday

- 6 years, 1 month ago

Ah I understand we'll wait.

- 6 years, 1 month ago

hmmm... I was wondering. is π a mathematical way of saying x?

- 2 months ago

Guys and Sirs Here is the OPC 4 !

- 6 years, 2 months ago

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

- 6 years, 1 month ago

Point 1 - I am not sir. (Just 14years old).
And for the or question U have to choose one of them. $\ddot\smile$

- 6 years, 1 month 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?

- 6 years, 1 month ago

Thanks. Keep on trying they are very easy. Especially 8,4,51,2(or).
Just curious : - Which one have U done ?

- 6 years, 1 month ago

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

- 6 years, 1 month ago

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

- 6 years, 1 month ago

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

- 6 years, 1 month ago

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

- 6 years, 1 month ago

^True dat.

- 6 years, 1 month ago

@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.

- 6 years, 1 month ago

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.

- 6 years, 1 month ago

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

- 6 years, 1 month ago

I'm still waiting. :/

- 6 years, 1 month ago

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

- 6 years, 1 month ago

Busy, will open when free.

- 6 years, 1 month ago

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

- 6 years, 1 month ago

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

- 6 years, 1 month ago

@Rajdeep Dhingra Dude.

- 6 years, 1 month ago

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

- 6 years, 2 months ago

Wait I will look into it. I think its AP

- 6 years, 2 months ago

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

- 6 years, 1 month ago

Thanks I will change it.

- 6 years, 1 month ago

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

- 6 years, 1 month ago

I check the inbox every afternoon so U will get the reply as soon as I check it. $\ddot\smile$

- 6 years, 1 month ago

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?

- 6 years, 1 month ago

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. $\ddot\smile$

- 6 years, 1 month ago