MathDB

10

Part of 2005 Harvard-MIT Mathematics Tournament

Problems(3)

2005 Algebra #10: Absolute Values of Roots

Source:

4/29/2013
Find the sum of the absolute values of the roots of x4−4x3−4x2+16x−8=0x^4 - 4x^3 - 4x^2 + 16x - 8 = 0.
quadraticsalgebraquadratic formula
2005 Calculus #10: Smooth Function at 1

Source:

4/29/2013
Let f:R→R f : \mathbf{R} \to \mathbf{R} be a smooth function such that f′(x)=f(1−x)f'(x)=f(1-x) for all xx and f(0)=1f(0)=1. Find f(1)f(1).
calculusfunction
2005 Geometry #10: Circle Traffic

Source:

5/16/2013
Let ABAB be a diameter of a semicircle Γ\Gamma. Two circles, ω1\omega_1 and ω2\omega_2, externally tangent to each other and internally tangent to Γ\Gamma, are tangent to the line ABAB at PP and QQ, respectively, and to semicircular arc ABAB at CC and DD, respectively, with AP<AQAP<AQ. Suppose FF lies on Γ\Gamma such that ∠FQB=∠CQA \angle FQB = \angle CQA and that ∠ABF=80∘ \angle ABF = 80^\circ . Find ∠PDQ \angle PDQ in degrees.
geometrygeometric transformationreflectiondilationcircumcirclepower of a pointradical axis