MathDB

1

Part of 2003 Hungary-Israel Binational

Problems(2)

n positive reals

Source: 16-th Hungary-Israel Binational Mathematical Competition 2003

3/30/2007
If x1,x2,...,xnx_{1}, x_{2}, . . . , x_{n} are positive numbers, prove the inequality x13x12+x1x2+x22+x23x22+x2x3+x32+...+xn3xn2+xnx1+x12≥x1+x2+...+xn3\frac{x_{1}^{3}}{x_{1}^{2}+x_{1}x_{2}+x_{2}^{2}}+\frac{x_{2}^{3}}{x_{2}^{2}+x_{2}x_{3}+x_{3}^{2}}+...+\frac{x_{n}^{3}}{x_{n}^{2}+x_{n}x_{1}+x_{1}^{2}}\geq\frac{x_{1}+x_{2}+...+x_{n}}{3}.
inequalitiesinequalities proposed
alternately write divisors of 100! on the blackboard

Source: 16-th Hungary-Israel Binational Mathematical Competition 2003

3/30/2007
Two players play the following game. They alternately write divisors of 100!100! on the blackboard, not repeating any of the numbers written before. The player after whose move the greatest common divisor of the written numbers equals 1,1, loses the game. Which player has a winning strategy?
greatest common divisorcombinatorics unsolvedcombinatorics