{"status": "success", "data": {"description_md": "In $\\triangle ABC, AB = 3$, $BC = 5$, and $AC = 7$. Let the incircle of $ABC$ be $\\omega _0$. Circle $\\omega_1$ is drawn inside $\\triangle ABC$ and outside $\\omega_0$, such that $\\omega_1$ is tangent to $\\omega_0$, $BC$, and $AC$. If the radius of $\\omega_1$ can be expressed as $\\frac{a\\sqrt b-c\\sqrt d}{e}$, where $a,b,c,d,e \\in \\mathbb{Z}^+$, $b$ and $d$ are not divisible by the square of any prime, and $\\gcd(a,c,e)=1$, find $a+b+c+d+e$.", "description_html": "

In \\triangle ABC, AB = 3, BC = 5, and AC = 7. Let the incircle of ABC be \\omega _0. Circle \\omega_1 is drawn inside \\triangle ABC and outside \\omega_0, such that \\omega_1 is tangent to \\omega_0, BC, and AC. If the radius of \\omega_1 can be expressed as \\frac{a\\sqrt b-c\\sqrt d}{e}, where a,b,c,d,e \\in \\mathbb{Z}^+, b and d are not divisible by the square of any prime, and \\gcd(a,c,e)=1, find a+b+c+d+e.

", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "TxO Math Bowl 2024 - Individuals A - Problem 10", "can_next": true, "can_prev": true, "nxt": "/problem/txo2024indivsA-p11", "prev": "/problem/txo2024indivsA-p09"}}