{"status": "success", "data": {"description_md": "Two quadratics $p(x)$ and $q(x)$ have all of their coefficients selected from the set $\\{1,2,3,\\ldots,10\\}$ such that $p(a) \\equiv q(a) \\pmod{3}$ for all integers $a$. Given that $p(x)-q(x)$ has exactly one real root, how many pairs $(p(x), q(x))$ exist?", "description_html": "<p>Two quadratics <span class=\"katex--inline\">p(x)</span> and <span class=\"katex--inline\">q(x)</span> have all of their coefficients selected from the set <span class=\"katex--inline\">\\{1,2,3,\\ldots,10\\}</span> such that <span class=\"katex--inline\">p(a) \\equiv q(a) \\pmod{3}</span> for all integers <span class=\"katex--inline\">a</span>. Given that <span class=\"katex--inline\">p(x)-q(x)</span> has exactly one real root, how many pairs <span class=\"katex--inline\">(p(x), q(x))</span> exist?</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 6, "problem_name": "TxO Math Bowl 2024 - Team Contest - Problem 19", "can_next": true, "can_prev": true, "nxt": "/problem/txo2024team-p20", "prev": "/problem/txo2024team-p18"}}