{"status": "success", "data": {"description_md": "Let $a, b, c, d$ be distinct integers in the set $\\{1,2, \\ldots, 9,10\\}$. They satisfy the following equations:\n\n$$a + c + d = b $$\n\n$$c = 2a $$\n\n$$a \\equiv 0 \\pmod 3 $$\n\nIf the sum of all rational roots of the equation $ax^3 + bx^2 + cx + d = 0$ can be expressed as $-\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find the value of $m + n$.", "description_html": "<p>Let <span class=\"katex--inline\">a, b, c, d</span> be distinct integers in the set <span class=\"katex--inline\">\\{1,2, \\ldots, 9,10\\}</span>. They satisfy the following equations:</p>&#10;<p><span class=\"katex--display\">a + c + d = b</span></p>&#10;<p><span class=\"katex--display\">c = 2a</span></p>&#10;<p><span class=\"katex--display\">a \\equiv 0 \\pmod 3</span></p>&#10;<p>If the sum of all rational roots of the equation <span class=\"katex--inline\">ax^3 + bx^2 + cx + d = 0</span> can be expressed as <span class=\"katex--inline\">-\\frac{m}{n}</span>, where <span class=\"katex--inline\">m</span> and <span class=\"katex--inline\">n</span> are relatively prime positive integers, find the value of <span class=\"katex--inline\">m + n</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 1, "problem_name": "TxO Math Bowl 2024 - Individuals B - Problem 4", "can_next": true, "can_prev": true, "nxt": "/problem/txo2024indivsB-p05", "prev": "/problem/txo2024indivsB-p03"}}