{"status": "success", "data": {"description_md": "Munch wants to scam people in a carnival game! However, all the people are smart, so he has to make the game fair (by fair, the expected amount of money Munch makes is $0$). Jerry, being bored, decides to play the game. The game is played as shown below:\n\n$\\qquad$****1.**** Jerry chooses a polygon (or line) $\\Gamma$.\n$\\qquad$****2.**** Jerry pays some amount of money to Munch.\n$\\qquad$****3.**** Jerry throws the polygon onto an infinite square grid of horizontal lines spaced $1$ unit apart and vertical lines spaced $1$ unit apart.\n$\\qquad$****4.**** Munch pays Jerry the number of intersections of $\\Gamma$ with the lines on the grid.\n\nFirst, Jerry gives Munch an equilateral triangle with side length $1$, and pays $\\dfrac{6}{\\pi}$. Then, Jerry gives Munch a regular octagon, and pays $\\dfrac{32}{\\pi}$. If the area of the octagon is $K$, find $\\lfloor K \\rfloor$.", "description_html": "<p>Munch wants to scam people in a carnival game! However, all the people are smart, so he has to make the game fair (by fair, the expected amount of money Munch makes is <span class=\"katex--inline\">0</span>). Jerry, being bored, decides to play the game. The game is played as shown below:</p>&#10;<p><span class=\"katex--inline\">\\qquad</span><strong><strong>1.</strong></strong> Jerry chooses a polygon (or line) <span class=\"katex--inline\">\\Gamma</span>.<br/>&#10;<span class=\"katex--inline\">\\qquad</span><strong><strong>2.</strong></strong> Jerry pays some amount of money to Munch.<br/>&#10;<span class=\"katex--inline\">\\qquad</span><strong><strong>3.</strong></strong> Jerry throws the polygon onto an infinite square grid of horizontal lines spaced <span class=\"katex--inline\">1</span> unit apart and vertical lines spaced <span class=\"katex--inline\">1</span> unit apart.<br/>&#10;<span class=\"katex--inline\">\\qquad</span><strong><strong>4.</strong></strong> Munch pays Jerry the number of intersections of <span class=\"katex--inline\">\\Gamma</span> with the lines on the grid.</p>&#10;<p>First, Jerry gives Munch an equilateral triangle with side length <span class=\"katex--inline\">1</span>, and pays <span class=\"katex--inline\">\\dfrac{6}{\\pi}</span>. Then, Jerry gives Munch a regular octagon, and pays <span class=\"katex--inline\">\\dfrac{32}{\\pi}</span>. If the area of the octagon is <span class=\"katex--inline\">K</span>, find <span class=\"katex--inline\">\\lfloor K \\rfloor</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 7, "problem_name": "TxO Math Bowl 2024 - Guts Contest - Set 8 Problem 3", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}