{"status": "success", "data": {"description_md": "Real numbers $x$, $y$, $z$ satisfy the following conditions:\n\n$\\qquad$****1.**** $-1\\le x \\le y \\le z \\le 1$\n\n$\\qquad$****2.**** $x\\ge \\dfrac{1}{2},$ or $x+y\\ge \\dfrac{2}{3},$ or $x+y+z\\ge \\dfrac{3}{2}$\n\nLet $\\max(\\min(\\{y-x, z-y, 1-z\\}))=\\frac{p}{q}$, for relatively prime positive integers $p$ and $q$. Compute $p+q$.", "description_html": "<p>Real numbers <span class=\"katex--inline\">x</span>, <span class=\"katex--inline\">y</span>, <span class=\"katex--inline\">z</span> satisfy the following conditions:</p>&#10;<p><span class=\"katex--inline\">\\qquad</span><strong><strong>1.</strong></strong> <span class=\"katex--inline\">-1\\le x \\le y \\le z \\le 1</span></p>&#10;<p><span class=\"katex--inline\">\\qquad</span><strong><strong>2.</strong></strong> <span class=\"katex--inline\">x\\ge \\dfrac{1}{2},</span> or <span class=\"katex--inline\">x+y\\ge \\dfrac{2}{3},</span> or <span class=\"katex--inline\">x+y+z\\ge \\dfrac{3}{2}</span></p>&#10;<p>Let <span class=\"katex--inline\">\\max(\\min(\\{y-x, z-y, 1-z\\}))=\\frac{p}{q}</span>, for relatively prime positive integers <span class=\"katex--inline\">p</span> and <span class=\"katex--inline\">q</span>. Compute <span class=\"katex--inline\">p+q</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 7, "problem_name": "TxO Math Bowl 2024 - Guts Contest - Set 8 Problem 2", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}