Consider $n$ circles with intersection by any two of them. Any area is all the common part between $m$ circles(a $m$-area): We have $2^n - 1$ areas, $m$ varies between $1$ and $m$.
A $1$-area is an area in a circle with no intersection with the other $n-1$ circles. A place is any surrounded part you can find, a place is different from an area.
The problem is this: Try to put integers in all the places with conditions below: The number in a $1$-area must be less than or equal to $0$, sum of the numbers in any $m$-area can be $0$ or $1$, no place can be empty. What is the maximum sum of the numbers you put in all the places in terms of $n$? Can it be more than $n/2$?!
Yes, it can be more than $n/2$. I have a counterexample.
Consider three circles, of which each pair intersects each other, but without common intersection.
Any two of them have an intersection, but there are six area's, this is less $2^3-1$, so you already made a wrong assumption here, or you forgot a requirement.
Now, put 0's in the 1-area's, and 1's in the 2-area's. All criteria related to your numbers are satisfied. The sum of the numbers is 3, which is clearly greater than 3/2.