Open Proof Contest 4 - Questions Needed

A intermediate problem by me ,

Q 1) Find all '$a$' such that $24\ | \ a^2+34a+1919$

1) Prove that

$\displaystyle\int_0^{\infty} \ dx \ln^2 \tanh (x) =\dfrac {7}{4}\zeta {(3) }$

$\tanh(x)$ is the hypebolic trigonometry function

You will be provided the solution if you want it. Cheers!

A intermediate problem by me,

Q1) On a chess-board if two squares are chosen, what is the probability that they have side in common .

You want algebra problem?

Let $a,b,c$ be positive reals satisfying $a^3+b^3+c^3+abc=4$ . Prove that $\frac{(5a^2+bc)^2}{(a+b)(a+c)} + \frac{(5b^2+ca)^2}{(b+c)(b+a)} + \frac{(5c^2+ab)^2}{(c+a)(c+b)} \ge \frac{(a^3+b^3+c^3+6)^2}{a+b+c}$ .

BONUS : How many cases of equality are there?

A slightly-above-intermediate problem by me.

Q #1 .

$\quad$

Find all integral solutions to the following equation:

$\quad$

$\frac {1}{m} + \frac{1}{n} - \frac {1} { mn^2} = \frac {3}{4}$

An easy proof problem :

Prove that $n!$ > $2^{n}$ for all integers $n \geq 4$.

This is a nice question but not original :

Q #2) $n$ 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 $n$.

Lemme post a geometry problem :

In $\Delta ABC$ , if $BC^2=AC(AC+AB)$ , then prove that $m\angle CAB = 2 \times m\angle CBA$.

@Sandeep Bhardwaj @Calvin Lin @Brian Charlesworth @Pi Han Goh

Respected sirs,
Please give questions.

@Archit Boobna @Parth Bhardwaj see this