{"status": "success", "data": {"description_md": "Let $\\ell_1$ and $\\ell_2$ be perpendicular lines intersecting at $O$. Let $A$, $B$, $C$ be different points on $\\ell_2$, appearing in that order and all on the same side of $O$. Let $OA=10$, $OB=26$, and $OC=35$.\n\nLet $\\omega_1$ be the circle passing through $A$ and $B$ tangent to $\\ell_1$. Let $\\omega_2$ be the circle, whose center is on the opposite side than that of $\\omega_1$ from $\\ell_2$, that passes through $A$ and $C$ and tangent to $\\ell_1$.\n\nFind the length of the common chord of the two circles, the answer is an integer.", "description_html": "<p>Let <span class=\"katex--inline\">\\ell_1</span> and <span class=\"katex--inline\">\\ell_2</span> be perpendicular lines intersecting at <span class=\"katex--inline\">O</span>. Let <span class=\"katex--inline\">A</span>, <span class=\"katex--inline\">B</span>, <span class=\"katex--inline\">C</span> be different points on <span class=\"katex--inline\">\\ell_2</span>, appearing in that order and all on the same side of <span class=\"katex--inline\">O</span>. Let <span class=\"katex--inline\">OA=10</span>, <span class=\"katex--inline\">OB=26</span>, and <span class=\"katex--inline\">OC=35</span>.</p>&#10;<p>Let <span class=\"katex--inline\">\\omega_1</span> be the circle passing through <span class=\"katex--inline\">A</span> and <span class=\"katex--inline\">B</span> tangent to <span class=\"katex--inline\">\\ell_1</span>. Let <span class=\"katex--inline\">\\omega_2</span> be the circle, whose center is on the opposite side than that of <span class=\"katex--inline\">\\omega_1</span> from <span class=\"katex--inline\">\\ell_2</span>, that passes through <span class=\"katex--inline\">A</span> and <span class=\"katex--inline\">C</span> and tangent to <span class=\"katex--inline\">\\ell_1</span>.</p>&#10;<p>Find the length of the common chord of the two circles, the answer is an integer.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 7, "problem_name": "Christmas Contest - Team Round - Problem 30", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/christmas1_team-p29"}}