Task B-3.2. The lengths of the sides of a triangle are three consecutive odd numbers, and the measure of one of the angles is $\frac{2 \pi}{3}$. Calculate the lengths of the sides of this triangle.
## Solution.

Let $a=2 n-1, b=2 n+1, c=2 n+3$.

Since $\frac{2 \pi}{3}$ is an obtuse angle, its opposite side is the longest, which is $c=2 n+3$.

From the cosine rule, we get

$$
\begin{aligned}
& \cos \frac{2 \pi}{3}=\frac{(2 n-1)^{2}+(2 n+1)^{2}-(2 n+3)^{2}}{2(2 n-1)(2 n+1)} \\
& \frac{-1}{2}=\frac{4 n^{2}-12 n-7}{2\left(4 n^{2}-1\right)} \\
& 8 n^{2}-12 n-8=0 \\
& n_{1,2}=\frac{3 \pm 5}{4}
\end{aligned}
$$

Since $n>0$, the solution to the equation is $n=2$.