Math Problems : SMT 2011 Geometry
9:24 PM
Muhammad Yusuf
, Posted in
1. Triangle ABC has side lengths BC = 3, AC = 4, AB = 5. Let P be a point inside or on triangle ABC and let the lengths of the perpendiculars from P to BC, AC, AB be Da, Db, Dc respectively. Compute the minimum of Da + Db + Dc.
2. Pentagon ABCDE is inscribed in a circle of radius 1. If ∠DEA ≌ ∠EAB ≌ ∠ABC, m∠CAD = 600, and BC = 2DE, compute the area of ABCDE.
3. Let circle O have radius 5 with diameter AE. Point F is outside circle O such that lines FA and FE intersect circle O at points B and D, respectively. If FA = 10 and m∠FAE = 300, then the perimeter of quadrilateral ABDE can be expressed as a + b√2 + c√3 + d√6, where a, b, c, and d are rational.
Find a + b + c + d.
4. Let ABC be any triangle, and D, E, F be points on BC, CA, AB such that CD = 2BD, AE = 2CE and BF = 2AF. AD and BE intersect at X, BE and CF intersect at Y , and CF and AD intersect at Z. Find Area(△ABC) : Area(△XYZ) .
5. Let ABCD be a cyclic quadrilateral with AB = 6, BC = 12, CD = 3, and DA = 6. Let E, F be the intersection of lines AB and CD, lines AD and BC respectively. Find EF.
6. Two parallel lines l1 and l2 lie on a plane, distance d apart. On l1 there are an infinite number of points A1,A2,A3, ...., in that order, with AnAn+1 = 2 for all n. On l2 there are an infinite number of points B1,B2,B3, .... , in that order and in the same direction, satisfying BnBn+1 = 1 for all n. Given that A1B1 is perpendicular to both l1 and l2, express the sum
7. In a unit square ABCD, find the minimum of √2AP + BP + CP where P is a point inside ABCD.
8. We have a unit cube ABCDEFGH where ABCD is the top side and EFGH is the bottom side with E below A, F below B, and so on. Equilateral triangle BDG cuts out a circle from the cube's inscribed sphere. Find the area of the circle.
9. We have a circle O with radius 10 and four smaller circles O1,O2,O3,O4 of radius 1 which are internally tangent to O, with their tangent points to O in counterclockwise order. The small circles do not intersect each other. Among the two common external tangents of O1 and O2, let l12 be the one which separates O1 and O2 from the other two circles, and let the intersections of l12 and O be A1 and B2, with A1 denoting the point closer to O1. Define l23, l34, l41 and A2,A3,A4,B3,B4,B1 similarly. Suppose that the arcs A1B1, A2B2, and A3B3 have length π, 3π/2, and 5π/2 respectively. Find the arc length of A4B4.
10. Given a triangle ABC with BC = 5, AC = 7, and AB = 8, find the side length of the largest equilateral triangle PQR such that A, B, C lie on QR, RP, PQ respectively.
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