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Suppose there are $n$ positive real numbers such that their sum is $20$ and the product is strictly greater than $1$. What is the maximum possible value of $n$?

Let us assume two positive real numbers $a$ and $b$. According to the above condition $a+b=20$ and $ab \gt 1$ This would lead to $a \gt \frac{1}{b}$. How do I use this information in the equation $a+b=20$ to find $n$?

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You can certainly have $n=19$ since you could have $19$ numbers all greater than $1$ with sum $20$.

To see that $n=19$ is the largest possible value that suits the conditions, think about the arithmetic-geometric mean inequality.

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    But the question states it is real numbers. I can have rational and irrationals coming into play and hence my value of $n$ would be quite large, wouldn't it? – 2017-03-09
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    @lcycarus Because the numbers (reals) are required to be positive, the above argument should still be fine. The AM-GM inequality holds in this situation. – 2017-03-09