PUMaC 2020 · 几何（A 组） · 第 16 题

Geometry A

1. 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 ?
2. 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 .
3. 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 ?
4. 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 ?
5. 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 2 a 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.
6. 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 .
7. Let ABC be a triangle with sides AB = 34 , BC = 15 , AC = 35 and let Γ be the circle of smallest possible radius passing through A tangent to BC . Let the second intersections of Γ and sides AB, AC be the points X, Y . Let the ray XY intersect the circumcircle of the triangle p ABC at Z . If AZ = for relatively prime integers p and q, find p + q. q
8. A A A A is a cyclic quadrilateral inscribed in circle Ω , with side lengths A A = 28 , A A = 1 2 3 4 1 2 2 3 √ √ 12 3 , A A = 28 3 , and A A = 8 . Let X be the intersection of A A , A A . Now, for 3 4 4 1 1 3 2 4 i = 1 , 2 , 3 , 4 , let ω be the circle tangent to segments A X, A X, and Ω , where we take i i i +1 indices cyclically (mod 4) . Furthermore, for each i, say ω is tangent to A A at X , A A at i 1 3 i 2 4 Y , and Ω at T . Let P be the intersection of T X and T X , and P the intersection of T X i i 1 1 1 2 2 3 3 3 and T X . Let P be the intersection of T Y and T Y , and P the intersection of T Y and 4 4 2 2 2 3 3 4 1 1 T Y . Find the area of quadrilateral P P P P . 4 4 1 2 3 4 1