Question 83
In $\triangle ABC$, the incircle centered at $I$ touches sides $AB$ and $BC$ at $X$ and $Y$, respectively. Additionally, the area of quadrilateral $BXIY$ is $\frac{2}{5}$ of the area of $ABC$. Let $p$ be the smallest possible perimeter of a $\triangle ABC$ that meets these conditions and has integer side lengths. Find the smallest possible area of such a triangle with perimeter $p$.