{"status": "success", "data": {"description_md": "A set $\\mathcal{S}$ has $2024$ elements. Suppose that the number of tuples of sets $(\\mathcal{S_1}, \\mathcal{S_2}, \\ldots, \\mathcal{S_{15}})$ that satisfy the following conditions is $k$:\n\n- Each set $\\mathcal{S_i}$ is a subset of $\\mathcal{S}$.\n- For even $i$, $\\mathcal{S_i}$ contains both $\\mathcal{S_{i-1}}$ and $\\mathcal{S_{i+1}}$.\n\nFind the remainder when $k$ is divided by $100$.", "description_html": "<p>A set <span class=\"katex--inline\">\\mathcal{S}</span> has <span class=\"katex--inline\">2024</span> elements. Suppose that the number of tuples of sets <span class=\"katex--inline\">(\\mathcal{S_1}, \\mathcal{S_2}, \\ldots, \\mathcal{S_{15}})</span> that satisfy the following conditions is <span class=\"katex--inline\">k</span>:</p>&#10;<ul>&#10;<li>Each set <span class=\"katex--inline\">\\mathcal{S_i}</span> is a subset of <span class=\"katex--inline\">\\mathcal{S}</span>.</li>&#10;<li>For even <span class=\"katex--inline\">i</span>, <span class=\"katex--inline\">\\mathcal{S_i}</span> contains both <span class=\"katex--inline\">\\mathcal{S_{i-1}}</span> and <span class=\"katex--inline\">\\mathcal{S_{i+1}}</span>.</li>&#10;</ul>&#10;<p>Find the remainder when <span class=\"katex--inline\">k</span> is divided by <span class=\"katex--inline\">100</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 7, "problem_name": "TxO Math Bowl 2024 - Individuals A - Problem 15", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/txo2024indivsA-p14"}}