MathDB

2

Part of 2011 Moldova Team Selection Test

Problems(2)

System of equations

Source: Moldova TST 2011, day 2, problem 2

3/8/2011
Find all pairs of real number xx and yy which simultaneously satisfy the following 2 relations: x+y+4=12x+11yx2+y2x+y+4=\frac{12x+11y}{x^2+y^2}
y−x+3=11x−12yx2+y2y-x+3=\frac{11x-12y}{x^2+y^2}
quadraticsalgebrapolynomialcomplex numbersalgebra unsolved
Product equals to 1

Source: Moldova TST 2011, day 1, problem 2

3/8/2011
Let x1,x2,…,xnx_1, x_2, \ldots, x_n be real positive numbers such that x1⋅x2⋯xn=1x_1\cdot x_2\cdots x_n=1. Prove the inequality 1x1(x1+1)+1x2(x2+1)+⋯+1xn(xn+1)≥n2\frac1{x_1(x_1+1)}+\frac1{x_2(x_2+1)}+\cdots+\frac1{x_n(x_n+1)}\geq\frac n2
inequalitiesfunctioninequalities unsolved