{"status": "success", "data": {"description_md": "A polynomial $P(x)$ of degree $2024$ with real coefficients satisfies the following conditions:\n\n- $P(x)=(x-1)^{240}\\cdot Q(x)$ for a polynomial $Q(x)$ with real coefficients, such that $Q(1) \\neq 0$.\n\n- In $P(x)$, the coefficient of the $x^i$ term is equal to the coefficient of the $x^{2024-i}$ term for nonnegative integers $i <1012$.\n\nGiven that all the roots of $P(x)$ are positive reals, how many of them are greater than $1$?", "description_html": "<p>A polynomial <span class=\"katex--inline\">P(x)</span> of degree <span class=\"katex--inline\">2024</span> with real coefficients satisfies the following conditions:</p>&#10;<ul>&#10;<li>&#10;<p><span class=\"katex--inline\">P(x)=(x-1)^{240}\\cdot Q(x)</span> for a polynomial <span class=\"katex--inline\">Q(x)</span> with real coefficients, such that <span class=\"katex--inline\">Q(1) \\neq 0</span>.</p>&#10;</li>&#10;<li>&#10;<p>In <span class=\"katex--inline\">P(x)</span>, the coefficient of the <span class=\"katex--inline\">x^i</span> term is equal to the coefficient of the <span class=\"katex--inline\">x^{2024-i}</span> term for nonnegative integers <span class=\"katex--inline\">i &lt;1012</span>.</p>&#10;</li>&#10;</ul>&#10;<p>Given that all the roots of <span class=\"katex--inline\">P(x)</span> are positive reals, how many of them are greater than <span class=\"katex--inline\">1</span>?</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 6, "problem_name": "March Break 2024 - Problem 10", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}