Three students named João, Maria, and José took a test with 100 questions, and each of them answered exactly 60 of them correctly. A question is classified as difficult if only one student answered it correctly, and it is classified as easy if all three answered it correctly. We know that each of the 100 questions was solved by at least one student. Are there more difficult or easy questions? Additionally, determine the difference between the number of difficult and easy questions.
Solution

We will say that a question is medium if exactly two students solved it correctly. Let $x, y$ and $z$ be the number of easy, medium, and hard questions, respectively. Since there are exactly 100 questions, we have $x+y+z=100$. By summing the number of questions solved by the three, which is $3 \cdot 60=180$, we will count the easy questions three times, as they will be counted in all three students, the medium questions two times, as they will be counted in two students, and the hard questions only once, as they should be counted in the student who solved them. Consequently, $3 x+2 y+z=180$. Therefore, if we multiply the first equation by 2 and subtract this last equation from it, we get:

$$
\begin{aligned}
2(x+y+z)-(3 x+2 y+z) & =200-180 \\
z-x & =20
\end{aligned}
$$

Thus, there are more hard questions than easy ones, and the difference between them is 20 units.