We have $600$ students and $240$ lectures. Every student has to visit at least $80$ lectures. I have to show that at least $40$ lectures have been visited by at least $100$ students.
$600 \cdot80 = 48000$ total visits have to be made. Let's put $99$ students into each lecture, so $99 \cdot 240 =23901$. $48000 - 23901 > 40$ and we are done.
This seemed too easy for me, is this really it already?
The idea about using the visits is correct, but you have only shown that we may not only have 99 students in each lecture. Instead, suppose that the property does not hold. Then, $201$ lectures are limited to $99$ attendance, and the remaining 39 are limited by the $600$ total students, so we may have at most $600\times 39+201\times 99$ visits, which is less the $48,000$ total required.