10.1. Let $a, b, c$ be not necessarily distinct natural numbers such that the fractions $\frac{a+b}{c}, \frac{a+c}{b}, \frac{b+c}{a}$ are also natural numbers. Find all possible values of the sum $\frac{a+b}{c}+\frac{a+c}{b}+\frac{b+c}{a}$.
Answer: 6, 7 or 8.

Solution. Due to symmetry, we can assume that $a \leq b \leq c$. Then $a+b \leq 2c$, so the integrality of the fraction $\frac{a+b}{c}$ implies $a+b=2c$ or $a+b=c$. In the first case, $a=b=c$ and the sum of the three fractions in the condition is 6. Let's consider further the case $a+b=c$. From the integrality of the fraction $\frac{a+c}{b}$, it follows that $2a$ is divisible by $b$, hence either $a=b$, and then $c=2a$, or $2a=b$ and then $c=3a$. In the first of these cases, the sum of the three fractions in the condition is 7, and in the second case, it is 8.

Grading criteria. The constraints $a+b=2c$ or $a+b=c$ are derived: 1 point for each. Examples of $a, b, c$ are provided for one or two of the answers 6, 7, or 8: 1 point, and if for all three answers: 2 points.