In an exam the average marks of all the students is $30$ (out of $100$). We have to show that at least $20\%$ of the students have got at least $10$ (out of $100$).
I know that it can be shown by contradiction, but I'm having difficulty forming the proof.
Let $x$ be the number of students. Suppose less than $\frac15$ of the students got at least $10$. Then more than $\frac45$ of the students got less than $10$. Their total number of marks is thus less than $\frac45 x\times 10+\frac15 x\times 100=28x$. This contradicts the fact that the total number of marks is $30x$.