The average of 20 numbers is 0. At most, how many of these numbers can be greater than 0?

Introduction / Context:An average of zero means the sum of all numbers is zero. We are asked for the maximum possible count of positive numbers under this constraint, with no restriction (other than arithmetic) on the negative numbers. This is a classic extremal question about sums.

Given Data / Assumptions:

• There are 20 real numbers (no further restrictions).
• Their sum must be 0 (average 0).

Concept / Approach:To maximize the count of positive numbers while keeping the sum at 0, we can take 19 of them to be any positive values and choose the remaining one number negative with magnitude equal to the sum of the 19 positives. This construction satisfies the sum constraint and uses as many positive entries as possible.

Step-by-Step Solution:

Verification / Alternative check:If all 20 were positive, the sum could not be zero. Hence 20 is impossible; 19 is attainable by construction, so it is the maximum.

Why Other Options Are Wrong:

• 0 or 1 are not maxima; much larger positive counts are possible.
• -1 is meaningless as a count of elements.

Final Answer:none of these (the correct maximum is 19).