HMMT 二月 2022 · 几何 · 第 6 题

HMMT February 2022 — Geometry — Problem 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 .

解析

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 . Proposed by: Ankit Bisain Answer: 21055 Solution: We have ( P A · P B )( P D · P C ) 2 · 18 P D · P A = = = 4 . ( P B · P C ) 9 ◦ ◦ Let α = ∠ DP C = 180 − ∠ AP B and β = ∠ AP D = ∠ BP C . Note that α + β = 90 . We have, letting x = AB = CD and y = AD = BC , 2[ P AD ] + 2[ P BC ] = y ( d ( P, AD ) + d ( P, BC )) = y · x = [ ABCD ] . Here d ( X, ℓ ) is used to denote the distance from X to line ℓ . By the trig area formula, the left-hand side is P A · P D · sin β + P B · P C · sin β = 13 sin β. Similarly, we have [ ABCD ] = 16 sin α . Thus, letting K = [ ABCD ], 2 2 K K 425 2 2 2 1 = sin α + sin β = + = K 2 2 2 2 13 16 13 · 16 √ 208 208 17 √ giving K = = . 85 425