{"status": "success", "data": {"description_md": "**POTD August 24, 2024**\n\nLet $[n] = \\{0,1,2,\\ldots,n\\}$. Call a non empty set $B \\subseteq [n]$ a block if $B$ is of the from $\\{i,i+1,\\ldots,j\\}$, for $i \\le j$. Let $S$ be the set of all blocks in $[2024]$. Say we can partition $S$ into disjoint subsets $S_1,S_2,\\ldots,S_k$ such that for any distinct $A,B \\in S_i$, we have $A \\cap B = \\varnothing$. What is the minimum value of $k$?", "description_html": "<p><strong>POTD August 24, 2024</strong></p>&#10;<p>Let <span class=\"katex--inline\">[n] = \\{0,1,2,\\ldots,n\\}</span>. Call a non empty set <span class=\"katex--inline\">B \\subseteq [n]</span> a block if <span class=\"katex--inline\">B</span> is of the from <span class=\"katex--inline\">\\{i,i+1,\\ldots,j\\}</span>, for <span class=\"katex--inline\">i \\le j</span>. Let <span class=\"katex--inline\">S</span> be the set of all blocks in <span class=\"katex--inline\">[2024]</span>. Say we can partition <span class=\"katex--inline\">S</span> into disjoint subsets <span class=\"katex--inline\">S_1,S_2,\\ldots,S_k</span> such that for any distinct <span class=\"katex--inline\">A,B \\in S_i</span>, we have <span class=\"katex--inline\">A \\cap B = \\varnothing</span>. What is the minimum value of <span class=\"katex--inline\">k</span>?</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 8, "problem_name": "Problem of the Day #255", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}