Problem 9.3. Four cities and five roads are arranged as shown in the figure. The lengths of all roads are equal to an integer number of kilometers. The lengths of four roads are indicated in the figure. How many kilometers is the length of the remaining one?

![](https://cdn.mathpix.com/cropped/2024_05_06_20e437e0605a873909d6g-2.jpg?height=488&width=638&top_left_y=87&top_left_x=412)

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# Answer: 17.

Solution. We will use the triangle inequality: in any non-degenerate triangle, the sum of any two sides is strictly greater than the remaining one.

Let $x$ km be the unknown length. From the left triangle, we see that $x < 10 + 8 = 18$. But if $x \leqslant 16$, then in the right triangle, the triangle inequality does not hold: $x + 5 \leqslant 21$. Since $x$ is an integer less than 18 and greater than 16, it equals 17.