Five hundred real numbers have an average of zero. How many of them can be negative, at most?
499
The average of 500 numbers is zero, which means their total sum is \(500\times 0 = 0\).
We want to make as many of the 500 numbers negative as we can while keeping the total at 0.
If every number were negative, the sum would be negative, not zero, so that is impossible.
To pull the sum back up to zero, at least one number must be positive and large enough to cancel all the negatives.
So we can make at most \(500 - 1 = 499\) numbers negative, keeping one positive.
For example, 499 numbers each equal to \(-1\) give a sum of \(-499\), and a single number \(+499\) balances it to exactly 0.
Key concept: a zero sum forces at least one term of opposite sign, so the maximum of one sign is total count minus one.
At most 499 of them can be negative.
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