{"status": "success", "data": {"description_md": "**POTD February 26, 2024**\n\nConsider an integer $0 \\le n \\le 127$. We define $3$ operations:\n\n**1.** If $n \\le 63$, we can replace $n$ with $2n$.\n\n**2.** If $n \\ge 64$, we may replace $n$ with $2n - 127$.\n\n**3.** If $95 \\ge n \\ge 64$, we may replace $n$ with $n - 32$.\n\nDefine $f(n)$ as the number of integers $n$ can reach with $0$ or more operations. Find the sum of all $f(n)$ across all $0 \\le n \\le 127$.", "description_html": "<p><strong>POTD February 26, 2024</strong></p>&#10;<p>Consider an integer <span class=\"katex--inline\">0 \\le n \\le 127</span>. We define <span class=\"katex--inline\">3</span> operations:</p>&#10;<p><strong>1.</strong> If <span class=\"katex--inline\">n \\le 63</span>, we can replace <span class=\"katex--inline\">n</span> with <span class=\"katex--inline\">2n</span>.</p>&#10;<p><strong>2.</strong> If <span class=\"katex--inline\">n \\ge 64</span>, we may replace <span class=\"katex--inline\">n</span> with <span class=\"katex--inline\">2n - 127</span>.</p>&#10;<p><strong>3.</strong> If <span class=\"katex--inline\">95 \\ge n \\ge 64</span>, we may replace <span class=\"katex--inline\">n</span> with <span class=\"katex--inline\">n - 32</span>.</p>&#10;<p>Define <span class=\"katex--inline\">f(n)</span> as the number of integers <span class=\"katex--inline\">n</span> can reach with <span class=\"katex--inline\">0</span> or more operations. Find the sum of all <span class=\"katex--inline\">f(n)</span> across all <span class=\"katex--inline\">0 \\le n \\le 127</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 4, "problem_name": "Problem of the Day #80", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}