HMMT 二月 2022 · 几何 · 第 25 题

HMMT February 2022 February 19, 2022 Geometry Round ◦

1. Let ABC be a triangle with ∠ A = 60 . Line ℓ intersects segments AB and AC and splits triangle ABC into an equilateral triangle and a quadrilateral. Let X and Y be on ℓ such that lines BX and CY are perpendicular to ℓ . Given that AB = 20 and AC = 22, compute XY .
2. Rectangle R has sides of lengths 3 and 4. Rectangles R , R , and R are formed such that: 0 1 2 3 • all four rectangles share a common vertex P , • for each n = 1 , 2 , 3, one side of R is a diagonal of R , n n − 1 • for each n = 1 , 2 , 3, the opposite side of R passes through a vertex of R such that the center n n − 1 of R is located counterclockwise of the center of R with respect to P . n n − 1 R 3 R 2 R 1 R 0 Compute the total area covered by the union of the four rectangles. 20
3. Let ABCD and AEF G be unit squares such that the area of their intersection is . Given that 21 ◦ a ∠ BAE < 45 , tan ∠ BAE can be expressed as for relatively prime positive integers a and b . Compute b 100 a + b .
4. Parallel lines ℓ , ℓ , ℓ , ℓ are evenly spaced in the plane, in that order. Square ABCD has the property 1 2 3 4 that A lies on ℓ and C lies on ℓ . Let P be a uniformly random point in the interior of ABCD and 1 4 let Q be a uniformly random point on the perimeter of ABCD . Given that the probability that P lies 53 a between ℓ and ℓ is , the probability that Q lies between ℓ and ℓ can be expressed as , where 2 3 2 3 100 b a and b are relatively prime positive integers. Compute 100 a + b .
5. Let triangle ABC be such that AB = AC = 22 and BC = 11. Point D is chosen in the interior of the ◦ 2 2 triangle such that AD = 19 and ∠ ABD + ∠ ACD = 90 . The value of BD + CD can be expressed a as , where a and b are relatively prime positive integers. Compute 100 a + b . b
6. Let ABCD be a rectangle inscribed in circle Γ, and let P be a point on minor arc AB of Γ. Suppose that P A · P B = 2, P C · P D = 18, and P B · P C = 9. The area of rectangle ABCD can be expressed √ a b as , where a and c are relatively prime positive integers and b is a squarefree positive integer. c Compute 100 a + 10 b + c .
7. Point P is located inside a square ABCD of side length 10. Let O , O , O , O be the circumcenters 1 2 3 4 √ of P AB , P BC , P CD , and P DA , respectively. Given that P A + P B + P C + P D = 23 2 and the area p a of O O O O is 50, the second largest of the lengths O O , O O , O O , O O can be written as , 1 2 3 4 1 2 2 3 3 4 4 1 b where a and b are relatively prime positive integers. Compute 100 a + b .
8. Let E be an ellipse with foci A and B . Suppose there exists a parabola P such that • P passes through A and B , • the focus F of P lies on E , • the orthocenter H of △ F AB lies on the directrix of P .

2 2 If the major and minor axes of E have lengths 50 and 14, respectively, compute AH + BH . 9. Let A B C , A B C , and A B C be three triangles in the plane. For 1 ≤ i ≤ 3, let D , E , and 1 1 1 2 2 2 3 3 3 i i F be the midpoints of B C , A C , and A B , respectively. Furthermore, for 1 ≤ i ≤ 3 let G be the i i i i i i i i centroid of A B C . i i i Suppose that the areas of the triangles A A A , B B B , C C C , D D D , E E E , and F F F 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 are 2, 3, 4, 20, 21, and 2020, respectively. Compute the largest possible area of G G G . 1 2 3 10. Suppose ω is a circle centered at O with radius 8. Let AC and BD be perpendicular chords of ω . Let P be a point inside quadrilateral ABCD such that the circumcircles of triangles ABP and CDP are √ √ tangent, and the circumcircles of triangles ADP and BCP are tangent. If AC = 2 61 and BD = 6 7, √ √ then OP can be expressed as a − b for positive integers a and b . Compute 100 a + b .