The average of 25 integers is zero. What is the maximum possible number of positive integers?

For the average of 25 integers to be zero, their sum must be zero:
x
1
​
+x
2
​
+⋯+x
25
​
=0
To maximize the number of positive integers, we can make 24 of the integers positive. The last (25
th
) integer can be a single large negative number equal to the sum of all the 24 positive integers, balancing the total sum to zero.

Hence, the maximum possible number of positive integers is 24.