Question 79
Let $n \geq 1$ be an integer. Ruben takes a test with $n$ questions. Every question on this test is worth a different number of points. The first question is worth 1 point, the second question 2, the third 3, and so on up to the last question, which is worth $n$ points. Every question is either right or wrong. So for a question he gets either no points or all the points. Let $f(n)$ be the number of ways in which he can take the test such that the number of points he obtained equals the number of questions he got wrong.
Investigate whether there exist infinitely many pairs $(a, b)$ with $a\lt b$ and $f(a)=f(b)$.
Investigate whether there exist infinitely many pairs $(a, b)$ with $a\lt b$ and $f(a)=f(b)$.