Simplify $2(x^8+y^8+z^8)-(x^4+y^4+z^4)^2$ (First Attempt)

Simplify $2(x^8+y^8+z^8)-(x^4+y^4+z^4)^2$.

Please don't be tempted by the temptation to expand the second term as it will lead to headache and no closer to the answer of the more simplified form:

Prove $\large 1000!^{\frac{1}{1000}}>999!^{\frac{1}{999}}$.

Prove $\large 1000!^{\frac{1}{1000}}>999!^{\frac{1}{999}}$.

This inequality would be easy to prove if one uses the more advance knowledge, like Stirling's formula where it states when $n\rightarrow \infty$ then we have $n!\approx \dfrac{n^n}{e^n}\sqrt{2\pi n}$.

But, elementary method works well too in this problem.

Hard Inequality Problem

Let $x,\,y,\,z$ be real numbers such that $6x+2y+3z=12+xyz$.

Prove that $(x^2+1)(y^2+9)(z^2+4)\ge 144$.

Note that

$(x^2+1)(y^2+9)(z^2+4)-(6x+2y+3z-xyz)^2$

Prove that: ⌊√(n)+√(n+1)⌋=⌊√(4n+2)⌋, for all positive integer n.

Prove that: $\left\lfloor{\sqrt{n}+\sqrt{n+1}}\right\rfloor= \left\lfloor{\sqrt{4n+2}}\right\rfloor$, for all $n\in N$.

My solution:

Step 1:

Note that:

$4n^2+4n\lt 4n^2+4n+1\lt 4n^2+8n+4$

Prove that: $⌊√n+1/(√n+√(n+2))⌋=⌊√n⌋$, for all $n\in N$.

Prove that: $\left\lfloor{\sqrt{n}+\dfrac{1}{\sqrt{n}+\sqrt{n+2}}}\right\rfloor= \left\lfloor{\sqrt{n}}\right\rfloor$, for all $n\in N$.

My solution:

Step 1:

First, note that the expression inside the floor function on the left can be rewritten such that we have:

Without the help of calculator, evaluate $\sqrt[8]{10828567056280801}$.

Without the help of calculator, evaluate $\sqrt[8]{10828567056280801}$.

This might look like there will be a lot of guessing before getting the right answer. But, if you toy around with the figure $10828567056280801$, it's not hard to see we could rewrite it so that we have:

$10828567056280801$

Prove $\sqrt{2}+\sqrt{3}\gt \pi$

Prove $\sqrt{2}+\sqrt{3}\gt \pi$.

Note that we could use the previously established result from the famous people or even our own finding to construct for the future conjectures and hence argument to help us in determining the plot that we are going to use to solve the new problem at hand, that is one of the very good traits of highly proficient problem solver.

Classic proof: $\dfrac{22}{7}\gt \pi$

Prove $\dfrac{22}{7}\gt \pi$.

There is a very classic and elegant proof for this inequality involving $\pi$ and its fraction representative $\dfrac{22}{7}$.

It's using the graphical and integration method to prove $\dfrac{22}{7}\gt \pi$.

Develep the power of observation

If you're given the two graphs that are plotted in the same Cartesian plane below, what do you notice?