How many zeros does a million and billion have. What is the equivalent lakhs and crore value of million and billion?

Interpreting z-scores: Complete the following statements using your knowledge about z-scores.
a. If the data is weight, the z-score for someone who is overweight would be
-positive
-negative
-zero
b. If the data is IQ test scores, an individual with a negative z-score would have a
-high IQ
-low IQ
-average IQ
c. If the data is time spent watching TV, an individual with a z-score of zero would
-watch very little TV
-watch a lot of TV
-watch the average amount of TV
d. If the data is annual salary in the U.S and the population is all legally employed people in the U.S., the z-scores of people who make minimum wage would be
-positive
-negative
-zero

A jetliner can fly 6.00 hours on a full load of fuel. Without wind it flies at a speed of 2.40 X 102m/s. The planes to make a round-trip by heading due west for a certain distance,turning around, and then heading due east for the return trip.During the entire flight, however, the place encounters a 57.8 m/s wind from the jet stream, which blows from west to east. What is the maximum distance that the plane can travel due west and just be able to return home ?

el diseño de un teatro consta de 27 sillas en la primera fila , 32 en la segunda, 37 en la tercera y asi sucesivamente. se sabe que el teatro tiene 10 filas.
¿ cuantas sillas tiene el teatro?

Olympiad Inequality $\sum _{cyc}\frac{x^4}{8x^3+5y^3}⩾\frac{x+y+z}{13}$
$x,y,z>0$ , prove$\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3}⩾\frac{x+y+z}{13}$
Note: Often Stack Exchange asked to show some work before answering the question. This inequality was used as a proposal problem for National TST of an Asian country a few years back. However, upon receiving the official solution, the committee decided to drop this problem immediately. They don't believe that any students can solve this problem in 3 hour time frame.
Update 1: In this forum, somebody said that BW is the only solution for this problem, which to the best of my knowledge is wrong. This problem is listed as "coffin problems" in my country. The official solution is very elementary and elegant.
Update 2: Although there are some solutions (or partial solution) based on numerical method, I am more interested in the approach with "pencil and papers." I think the approach by Peter Scholze in here may help.
Update 3: Michael has tried to apply Peter Scholze's method but not found the solution yet.
Update 4: Symbolic expanding with computer is employed and verify the inequality. However, detail solution that not involved computer has not been found. Whoever can solve this inequality using high school math knowledge will be considered as the "King of Inequality".

Consider the wave equation with initial data:
$u_{tt}(t,x)+u_{xx}(t,x)=0$
$u(0,x)=u_0(x)$
$u_t(0,x)=u_1(x)$
Hadamard showed that this problem is ill-posed: there exist large solutions with arbitrarily small initial data. For instance, if we take $u(t,x)=a_{\omega }\sinh (\omega t)\sin (\omega x)$, then $u_0(x)=0$ and $u_1(x)=a_{\omega }\omega \sin (\omega x)$, then we can make u(t,x) grow arbitrarily fast while keeping $u_0$ and $u_1$ small.
Tweaking this construction, it is not hard to see that for any given k and $ϵ>0$, we can construct initial data such that
$||u_0|{|}_{\mathrm{\infty }}+||u_0^{(1)}|{|}_{\mathrm{\infty }}+\dots +||u_0^{(k)}|{|}_{\mathrm{\infty }}+||u_1|{|}_{\mathrm{\infty }}+||u_1^{(1)}|{|}_{\mathrm{\infty }}+\dots +||u_1^{(k)}|{|}_{\mathrm{\infty }}<ϵ$
and $||u(ϵ,x)|{|}_{\mathrm{\infty }}>\frac{1}{ϵ}$This can be interpreted as saying that the problem is ill-posed even in a Holder sense.
My question is: can one construct an example of a solution u(t,x) with initial data $u_0(x)$ and $u_1(x)$ such that
$\sum _{i=0}^{\mathrm{\infty }}||u_0^{(i)}|{|}_{\mathrm{\infty }}+||u_1^{(i)}|{|}_{\mathrm{\infty }}<ϵ$

Sorry if this questions is overly simplistic. It's just something I haven't been able to figure out.
I've been reading through quite a few linear algebra books and have gone through the various methods of solving linear systems of equations, in particular, n systems in n unknowns. While I understand the techniques used to solve these for the most part, I don't understand how these situations present themselves. I was wondering if anyone could provide a simple real-world example or two from data analysis, finance, economics, etc. in which the problem they were working on led to a system of n equations in n unknowns. I don't need the solution worked out. I just need to know the problem that resulted in the system.

Below is data collected from the growth of two different trees over time. Each tree was planted in 1960, and the trees