MathDB

6

Part of 2007 Harvard-MIT Mathematics Tournament

Problems(4)

2007 Algebra #6: Complex Inputs, Real Outputs

Source:

6/21/2012
Consider the polynomial P(x)=x3+x2−x+2P(x)=x^3+x^2-x+2. Determine all real numbers rr for which there exists a complex number zz not in the reals such that P(z)=rP(z)=r.
algebrapolynomial
2007 Calculus #6: Tangent Elliptic Curve

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6/21/2012
The elliptic curve y2=x3+1y^2=x^3+1 is tangent to a circle centered at (4,0)(4,0) at the point (x0,y0)(x_0,y_0). Determine the sum of all possible values of x0x_0.
calculusfunctionderivative
2007 Geometry #6: Cevian intersecting Incircle

Source:

6/22/2012
Triangle ABCABC has ∠A=90∘\angle A=90^\circ, side BC=25BC=25, AB>ACAB>AC, and area 150150. Circle ω\omega is inscribed in ABCABC, with MM its point of tangency on ACAC. Line BMBM meets ω\omega a second time at point LL. Find the length of segment BLBL.
geometry
2007 Guts #6: Three Video Game Systems

Source:

6/22/2012
There are three video game systems: the Paystation, the WHAT, and the ZBoz2π\pi, and none of these systems will play games for the other systems. Uncle Riemann has three nephews: Bernoulli, Galois, and Dirac. Bernoulli owns a Paystation and a WHAT, Galois owns a WHAT and a ZBoz2π\pi, and Dirac owns a ZBoz2π\pi and a Paystation. A store sells 44 different games for the Paystation, 66 different games for the WHAT, and 1010 different games for the ZBoz2π\pi. Uncle Riemann does not understand the difference between the systems, so he walks into the store and buys 33 random games (not necessarily distinct) and randomly hands them to his nephews. What is the probability that each nephew receives a game he can play?
videosprobability