The average of 20 numbers is zero. Of them, at the most, how many may be…

Concept

For any finite collection, the arithmetic mean equals the sum of all values divided by the number of values. Therefore, a zero mean requires a total sum of zero. If some values are positive, their positive total must be balanced by the remaining values.

Application

1. Let p be the number of positive values among the 20 numbers.

2. If p = 20, every value is positive, so their sum must be positive; it cannot be zero. Hence p ≤ 19.

3. The bound is attainable: choose nineteen values equal to 1 and the remaining value equal to −19. Their sum is 19 × 1 + (−19) = 0, so the mean is 0.

Cross-check

Nineteen positive values are possible, while twenty positive values are impossible. Therefore, the greatest possible number of values greater than zero is 19.