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Omni-MATH / In △ABC, the incircle centered at I touches sides AB and BC at X and Y, respectively. Additionally, the area of quadrilateral BXIY is (2)/(5) of the area of ABC. Let p be the small…

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problem

In △ABC\triangle A B C, the incircle centered at II touches sides ABA B and BCB C at XX and YY, respectively. Additionally, the area of quadrilateral BXIYB X I Y is 25\frac{2}{5} of the area of ABCA B C. Let pp be the smallest possible perimeter of a △ABC\triangle A B C that meets these conditions and has integer side lengths. Find the smallest possible area of such a triangle with perimeter pp.
Plain-text mathematical notation (without MathML)
In △ABC, the incircle centered at I touches sides AB and BC at X and Y, respectively. Additionally, the area of quadrilateral BXIY is (2)/(5) of the area of ABC. Let p be the smallest possible perimeter of a △ABC that meets these conditions and has integer side lengths. Find the smallest possible area of such a triangle with perimeter p.
Original LaTeX notation
In $\triangle A B C$, the incircle centered at $I$ touches sides $A B$ and $B C$ at $X$ and $Y$, respectively. Additionally, the area of quadrilateral $B X I Y$ is $\frac{2}{5}$ of the area of $A B C$. Let $p$ be the smallest possible perimeter of a $\triangle A B C$ that meets these conditions and has integer side lengths. Find the smallest possible area of such a triangle with perimeter $p$.

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task_id
42002d31-4a67-5ccf-a9f6-ebc15e57ed65
task_key
test--42002d31-4a67-5ccf-a9f6-ebc15e57ed65
task_revision_id
1
config / split
/ test

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