PUMaC 2020 · 几何（B 组） · 第 16 题

Geometry B

1. You are walking along a road of constant width with sidewalks on each side. You can only walk on the sidewalks or cross the road perpendicular to the sidewalk. Coming up on a turn, you realize that you are on the “outside” of the turn; i.e., you are taking the longer way around the turn. The turn is a circular arc. Assuming that your destination is on the same side of the road as you are currently, let θ be the smallest turn angle, in radians, that would justify crossing the road and then crossing back after the turn to take the shorter total path to your destination. What is b 100 × θ c ?
2. Seven students in Princeton Juggling Club are searching for a room to meet in. However, they must stay at least 6 feet apart from each other, and due to midterms, the only open rooms they can find are circular. In feet, what is the smallest diameter of any circle which can contain seven points, all of which are at least 6 feet apart from each other?
3. Let γ and γ be circles centered at O and P respectively, and externally tangent to each other 1 2 at point Q. Draw point D on γ and point E on γ such that line DE is tangent to both 1 2 circles. If the length OQ = 1 and the area of the quadrilateral ODEP is 520 , then what is the value of length P Q ?
4. Hexagon ABCDEF has an inscribed circle Ω that is tangent to each of its sides. If AB = 12, √ ◦ ◦ ∠ F AB = 120 , and ∠ ABC = 150 , and if the radius of Ω can be written as m + n for positive integers m, n , find m + n .
5. Let ABCD be a cyclic quadrilateral with circumcenter O and radius 10 . Let sides AB, BC, CD, and DA have midpoints M, N, P, and Q, respectively. If M P = N Q and OM + OP = 16 , then what is the area of triangle 4 OAB ?
6. Let C be a circle centered at point O, and let P be a point in the interior of C. Let Q be a point on the circumference of C such that P Q ⊥ OP, and let D be the circle with diameter P Q. Consider a circle tangent to C whose circumference passes through point P. Let the curve Γ be the locus of the centers of all such circles. If the area enclosed by Γ is 1 / 100 the area of C, then what is the ratio of the area of C to the area of D ?
7. Triangle ABC is so that AB = 15 , BC = 22 , and AC = 20 . Let D, E, F lie on BC, AC, and AB, respectively, so AD, BE, CF all contain a point K. Let L be the second intersection of AK 11 a 2 the circumcircles of BF K and CEK. Suppose that = , and BD = 6 . If KL = , where KD 7 b a, b are relatively prime integers, find a + b.
8. Triangle ABC has side lengths 13, 14, and 15. Let E be the ellipse that encloses the smallest √ a bπ area which passes through A , B , and C . The area of E is of the form , where a and c are c coprime and b has no square factors. Find a + b + c . 1