2000 PMWC Problems/Problem T8

Problem

There are positive integers $k$ , $n$ , $m$ such that $\frac{19}{20}<\frac{1}{k}+\frac{1}{n}+\frac{1}{m}<1$ . What is the smallest possible value of $k+n+m$ ?

Solution

Given that $k$ , $n$ , and $m$ are positive integers, it is obvious that none of them can be 1, because that would immediately make the quantity $\frac{1}{k}+\frac{1}{n}+\frac{1}{m}$ too large.

What about having one of them equal $2$ ? We can actually prove through casework that at least one of the integers must be equal to $2$ by considering what would happen if none of $k$ , $n$ , or $m$ were equal to $2$ . If this was true, then the smallest $k$ , $n$ , or $m$ could be would be $3$ . If all three were equal to $3$ , then we would have $\frac{1}{3}+\frac{1}{3}+\frac{1}{3} = 1$ , which is too large. However, the next largest value for the sum $\frac{1}{k}+\frac{1}{n}+\frac{1}{m}$ would be $\frac{1}{3}+\frac{1}{3}+\frac{1}{4} = \frac{11}{12}$ , which is too small. Thus there are no possibilities that have no $2$ s, meaning we can set any one of the variables $k$ , $n$ , or $m$ equal to $2$ and continue solving.

Let's say we choose $k$ to equal $2$ . Then we have $\frac{9}{20}<\frac{1}{n}+\frac{1}{m}<\frac{1}{2}$ . Obviously we can't have another $2$ , but what if we had a $3$ ? It turns out that we can prove that we must have a $3$ in much the same way we proved that there must be a $2$ . If we assume that there is no $3$ , the largest value of $\frac{1}{n}+\frac{1}{m}$ would be $\frac{1}{4}+\frac{1}{4} = \frac{1}{2}$ , which is too large. Just as last time, the next largest possibility, $\frac{1}{4}+\frac{1}{5} = \frac{9}{20}$ , is too small. Thus, there are no possibilities that have no $3$ s, meaning one of the remaining variables must be $3$ . If we randomly select $n$ to equal $3$ , we are left with the equation $\frac{9}{20} - \frac{1}{3} < \frac{1}{m} < \frac{1}{2} - \frac{1}{3}$ , which means $\frac{7}{60} < \frac{1}{m} < \frac{1}{6}$ . This inequality shows that $m$ can be equal to $7$ or $8$ . The problem asks for the least possible value of $k+n+m$ , so we say that $m = 7$ .

To conclude, we have $\frac{1}{k}+\frac{1}{n}+\frac{1}{m} = \frac{1}{2}+\frac{1}{3}+\frac{1}{7} = \frac{21}{42}+\frac{14}{42}+\frac{6}{42} = \frac{41}{42}$ , which is indeed greater than $\frac{19}{20}$ and less than $1$ . This configuration, $(2,3,7)$ is one of two that satisfies the given conditions, the other being $(2,3,8)$ . Of these two solutions, $(2,3,7)$ has the lower sum, meaning our answer is $2 + 3 + 7 = \boxed{12}$

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2000 PMWC Problems/Problem T8

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