MathDB

2

Part of 1995 China Team Selection Test

Problems(2)

locus of P as M moves on major arc AB of the larger circle

Source: China TST 1995, problem 2

5/17/2005
Given a fixed acute angle θ\theta and a pair of internally tangent circles, let the line ll which passes through the point of tangency, AA, cut the larger circle again at BB (ll does not pass through the centers of the circles). Let MM be a point on the major arc ABAB of the larger circle, NN the point where AMAM intersects the smaller circle, and PP the point on ray MBMB such that ∠MPN=θ\angle MPN = \theta. Find the locus of PP as MM moves on major arc ABAB of the larger circle.
geometrygeometric transformationgeometry unsolvedTangentsLocus problemsChinaTST
China TST 1995 polynomial game

Source: China TST 1995, problem 5

5/17/2005
A A and B B play the following game with a polynomial of degree at least 4: x^{2n} \plus{} \_x^{2n \minus{} 1} \plus{} \_x^{2n \minus{} 2} \plus{} \ldots \plus{} \_x \plus{} 1 \equal{} 0 A A and B B take turns to fill in one of the blanks with a real number until all the blanks are filled up. If the resulting polynomial has no real roots, A A wins. Otherwise, B B wins. If A A begins, which player has a winning strategy?
algebrapolynomialalgebra unsolved