We know that 60 students solved the fourth problem, and all of them are potential winners since they also solved the third problem. However, we need to subtract those who solved all four problems since they are not eligible to be winners.

Let’s call the number of students who solved all four problems “x”. Since no one managed to solve all four problems, we have x = 0.

Therefore, the number of winners is equal to the number of students who solved the third and fourth problems but not all four. This is equal to the intersection of the sets of students who solved the third problem and those who solved the fourth problem, minus the set of students who solved all four problems:

{students who solved the third problem} ∩ {students who solved the fourth problem}

-

{students who solved all four problems}

=

{students who solved the third problem and the fourth problem}

- 0

= 70 + 60 - 100

= 30

Therefore, there were 30 winners of the contest.