5. There are 100 integers placed on a circle. It is known that each integer is greater than the sum of the next two numbers in the clockwise direction. Question: What is the maximum number of positive integers among these 100?
5. 49 .

Obviously, there do not exist two non-negative numbers adjacent. Otherwise, the next number is positive, thereby making all numbers positive. But the smallest number among them cannot be greater than the sum of the next two numbers. This indicates that there are at most 50 non-negative numbers. If there are exactly 50 positive numbers, then these positive and negative numbers are alternately arranged.
Consider three adjacent numbers $-a, b, -c (a, b, c \geqslant 0)$.
Then $-a > b - c \geqslant -c$.
This indicates that the non-positive numbers are monotonically arranged, which is a contradiction.
Therefore, the number of positive numbers does not exceed 49.
An example with 49 positive numbers:
$$
\begin{array}{l}
-200, 1, -202, 1, -204, 1, -206, 1, \\
\cdots, -296, 1, -298, -99 .
\end{array}
$$