Given
- Average of 20 numbers = 0 ⇒ their sum = 0
Goal
- Maximize how many can be > 0 (positive).
Idea
- To keep the total sum at 0, positives must be balanced by non-positives (zeros and/or negatives).
- To maximize the count of positive numbers, make as many of the remaining numbers zero as possible and keep just one negative number whose magnitude equals the sum of the positives.
Construction
Take 19 positive numbers (any values), 0 zeros, and 1 negative number = −(sum of those 19 positives).Sum = 0 and 19 numbers are positive.Conclusion
At most 19 numbers may be greater than zero.
Discussion & Comments