Are these statistical statements sometimes, always or never true? Or it is impossible to say?

Given a probability density function find the mean, median and mode of the distribution.

What is the largest that the mean of these numbers could be?

Find the value of $m$ from these statements about a group of numbers

Can you make all of these statements about averages true at the same time?

How can people be divided into groups fairly for events in the Paralympics, for school sports days, or for subject sets?

Given the mean and standard deviation of a set of marks, what is the greatest number of candidates who could have scored 100%?

Can you decide whether these short statistical statements are always, sometimes or never true?

Kyle and his teacher disagree about his test score - who is right?

Charlie has moved between countries and the average income of both has increased. How can this be so?