1. The lengths of the sides of a triangle are $a=b-\frac{r}{4}, b, c=b+\frac{r}{4}$, where $r$ is the radius of the circle inscribed in the triangle. Express the lengths of the sides of the triangle in terms of $r$.


1. Let $s=\frac{a+b+c}{2}=\frac{3}{2} b$. From the formula for the area of a triangle, we get

$$
s r=P=\sqrt{s(s-a)(s-b)(s-c)}
$$

10 points

By substituting the expressions for $a, b$, and $c$ above, we have

$$
\begin{aligned}
s r^{2} & =(s-a)(s-b)(s-c) \\
\frac{3}{2} b r^{2} & =\left(\frac{b}{2}+\frac{1}{4} r\right) \frac{b}{2}\left(\frac{b}{2}-\frac{1}{4} r\right) \\
b^{2} & =\frac{49}{4} r^{2}
\end{aligned}
$$

from which $b=\frac{7}{2} r, a=\frac{13}{4} r$ and $c=\frac{15}{4} r$.

15 points