{"status": "success", "data": {"description_md": "Suppose that $PQRSTUVW$ is a regular octagon. (A *regular octagon* is an octagon with eight equal side lengths and eight equal interior angles.) There are 70 ways in which four of its sides can be chosen at random. If four of its sides are chosen at random and each of these sides is extended infinitely in both directions, what is the probability that they will meet to form a quadrilateral that contains the octagon?\n\n$\\text{(A)}\\ \\frac{1}{2}\\qquad\\text{(B)}\\ \\frac{19}{35}\\qquad\\text{(C)}\\ \\frac{37}{70}\\qquad\\text{(D)}\\ \\frac{17}{35}\\qquad\\text{(E)}\\ \\frac{18}{35}$\n\n___\n\n**[Cayley](https://www.cemc.uwaterloo.ca/contests/past_contests.html#pcf), by [CEMC](https://cemc.uwaterloo.ca/), is licensed under the [CC BY-NC 3.0 DEED](https://creativecommons.org/licenses/by-nc/3.0/). They are not affiliated with or endorse TopsOJ in any way.**\n", "description_html": "

Suppose that PQRSTUVW is a regular octagon. (A *regular octagon* is an octagon with eight equal side lengths and eight equal interior angles.) There are 70 ways in which four of its sides can be chosen at random. If four of its sides are chosen at random and each of these sides is extended infinitely in both directions, what is the probability that they will meet to form a quadrilateral that contains the octagon?

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\\text{(A)}\\ \\frac{1}{2}\\qquad\\text{(B)}\\ \\frac{19}{35}\\qquad\\text{(C)}\\ \\frac{37}{70}\\qquad\\text{(D)}\\ \\frac{17}{35}\\qquad\\text{(E)}\\ \\frac{18}{35}

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Cayley, by CEMC, is licensed under the CC BY-NC 3.0 DEED. They are not affiliated with or endorse TopsOJ in any way.

\n", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2016 Cayley Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/16_cayley_p24", "prev": "/problem/16_cayley_p22"}}