{"status": "success", "data": {"description_md": "For positive integers $a < b$, denote $$f(a,b) = 2\\ln\\left(\\prod_{k = a}^{b-1}\\left|\\dfrac{1+k+k^2+i}{1+k^2}\\right|\\right).$$ For a prime $p \\equiv 3 \\pmod{4}$, let $$\\ln(g(p)) = \\sum_{1 \\le a < b < p}f(a,b).$$ Find the sum of $g(p) \\pmod{p}$ across all primes $p \\le 100$ and $p \\equiv 3 \\pmod 4$, where $g(p) \\pmod{p}$ gives the integer $k$ such that $k \\equiv g(p) \\pmod{p}$ and $0 \\le k < p$.", "description_html": "<p>For positive integers <span class=\"katex--inline\">a &lt; b</span>, denote <span class=\"katex--display\">f(a,b) = 2\\ln\\left(\\prod_{k = a}^{b-1}\\left|\\dfrac{1+k+k^2+i}{1+k^2}\\right|\\right).</span> For a prime <span class=\"katex--inline\">p \\equiv 3 \\pmod{4}</span>, let <span class=\"katex--display\">\\ln(g(p)) = \\sum_{1 \\le a &lt; b &lt; p}f(a,b).</span> Find the sum of <span class=\"katex--inline\">g(p) \\pmod{p}</span> across all primes <span class=\"katex--inline\">p \\le 100</span> and <span class=\"katex--inline\">p \\equiv 3 \\pmod 4</span>, where <span class=\"katex--inline\">g(p) \\pmod{p}</span> gives the integer <span class=\"katex--inline\">k</span> such that <span class=\"katex--inline\">k \\equiv g(p) \\pmod{p}</span> and <span class=\"katex--inline\">0 \\le k &lt; p</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 7, "problem_name": "Holiday Contest 2025 - Individual Round - Problem 10", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}