MathDB

3

Part of 2000 Vietnam Team Selection Test

Problems(2)

game on polynomials

Source: Vietnam TST 2000

4/3/2007
Two players alternately replace the stars in the expression ∗x2000+∗x1999+...+∗x+1*x^{2000}+*x^{1999}+...+*x+1 by real numbers. The player who makes the last move loses if the resulting polynomial has a real root tt with ∣t∣<1|t| < 1, and wins otherwise. Give a winning strategy for one of the players.
algebrapolynomialalgebra proposed
A collection of $2000$ congruent circles

Source: Vietnam TST 2000

4/3/2007
A collection of 20002000 congruent circles is given on the plane such that no two circles are tangent and each circle meets at least two other circles. Let NN be the number of points that belong to at least two of the circles. Find the smallest possible value of NN.
combinatorics proposedcombinatorics