{"status": "success", "data": {"description_md": "A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\\sqrt{2}$ in the center of the square, as shown in the diagram. A circular coin with diameter $1$ is dropped onto the square and lands in a random location where the coin is completely contained within the square. The probability that the coin will cover part of the black region of the square can be written as $\\frac{1}{196}\\left(a+b\\sqrt{2}+\\pi\\right)$, where $a$ and $b$ are positive integers. What is $a+b$?\n\n
\n\n
\n\n$\\textbf{(A)} ~64 \\qquad\\textbf{(B)} ~66 \\qquad\\textbf{(C)} ~68 \\qquad\\textbf{(D)} ~70 \\qquad\\textbf{(E)} ~72$", "description_html": "

A square with side length 8 is colored white except for 4 black isosceles right triangular regions with legs of length 2 in each corner of the square and a black diamond with side length 2\\sqrt{2} in the center of the square, as shown in the diagram. A circular coin with diameter 1 is dropped onto the square and lands in a random location where the coin is completely contained within the square. The probability that the coin will cover part of the black region of the square can be written as \\frac{1}{196}\\left(a+b\\sqrt{2}+\\pi\\right) , where a and b are positive integers. What is a+b ?

\n
\n\n
\n

\\textbf{(A)} ~64 \\qquad\\textbf{(B)} ~66 \\qquad\\textbf{(C)} ~68 \\qquad\\textbf{(D)} ~70 \\qquad\\textbf{(E)} ~72

\n

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.

", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 4, "problem_name": "2021 AMC 10B Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/21_amc10B_p24", "prev": "/problem/21_amc10B_p22"}}