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