Can anyone help me with statistics? Question: Following is the distribution of marks obtained by 5

$\mu =m=\frac{q_1+q_3}{2}$
Let $X$ be a random variable with p.m.f./p.d.f. $f_X(x)$ that is symmetric about $\mu (\in R),$ i.e., $f_X(x+\mu )=f_X(\mu -x)$, $\mathrm{\forall }x\in (-\mathrm{\infty },\mathrm{\infty }).$
If $q_1,m$ and $q_3$ are respectively the lower quartile, the median and the upper quartile of the distribution of $X$ then show that $\mu =m=\frac{q_1+q_3}{2}$
How to prove.....
any Hints.

What is the meaning of percentile?
I am confused by the term percentile. Once my teacher told me that percentile means the percentage with respect to the score of the highest achiever.
This means that if in a competition I got $80$ out of $100$ and the highest score in that competition was 90 out of 100 then my percentile would be $\frac{80}{90}\ast 100=88.89$
So I got $80\mathrm{\%}$ and $88.89$ percentile.
I was believing that my above concept was right.
But when I see the definition of percentile on Wikipedia then I got something new (but I don't understand this definition) and then I thought that what my teacher told me was wrong.
Kindly tell me if my teacher right or wrong.

Calculating the position of the median
I know this is a simple question, but I cannot find a straight answer anywhere.
When calculating the medium of listed data, the formula is (n+1)/2. My Statistics teacher said for grouped data the position of the medium is n/2. However, this seems contradictory as for discrete grouped data, the data could be written in a list if the original values were known. Therefore two different values for the median are found.
I know this question is probably going to get flagged as it has already been asked, however, it has never been answered. I find it frustrating that such a fundamental concept in Statistics, what is supposed to be precise and never subjective has a wishy-washy answer.
Rant over, I think the position should be (n+1)/2 for discrete data and n/2 for continuous data. Moreover, this then leads into the obvious question of how we should calculate quartiles as well.
It would be nice if everyone could agree on a certain method for calculating the medium as all my textbooks are saying different things.
Someone please dispell the confusion and this statistical mess.

calculating upper and lower quartiles
The number of goals scored by a football team during a months worth of matches is recorderd.
They scored $0$ goals $4$ times, $1$ goal $6$ six times and $2$ goals $3$ times. Calculate the lower and upper quartiles.
Would it simply be this?: lower quartile is the $3.5$th value which is zero, and upper quartile is the $10.5$th value which is $2$?

What is the lower quartile of the set of data?
I came across this problem asking the lower quartile of the ungrouped data. My answer is 3, but other references say it should be 2.5. Here's the data:
1, 1, 2, 2, 3, 3, 4, 4, 6, 7, 8, 10, 11, 14, 15, 20, 22
What do you think?

Probability and Statistics (Quartiles)
Q: Find the upper and lower quartiles of the random variable.
$f(x)=\frac{1}{x\ln (1.5)}$
for $4\le x\le 6$
I set $F(x)=0.25$ and $F(x)=0.75$ to denote $Q1$ and $Q3$
This is what I got for the answers:
For
$F(X)=0.25,x=9.86$
For
$F(X)=0.75,x=3.28$
However, these answers are wrong as compared with my textbook. It should be:
For
$F(X)=0.25,x=4.43$
For
$F(X)=0.75,x=5.42$
Please help me. I don't know what I'm doing wrong.