5. (8 points) In $\triangle A B C, A B=86$, and $A C=97$. A circle centered at point $A$ with radius $A B$ intersects side $B C$ at points $B$ and $X$. Additionally, $B X$ and $C X$ have integer lengths. What is the length of $B C ?$
Answer: 61

Solution: Let $x=B X$ and $y=C X$. We will calculate the power of point $C$ in two ways

$$
y(y+x)=97^{2}-86^{2}=2013
$$

Considering all divisors of the number 2013 and taking into account the triangle inequality $\triangle A C X$, we obtain the unique solution 61.