MathDB

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Part of 2014 Middle European Mathematical Olympiad

Problems(2)

not so hard functional equation

Source: Middle European Mathematical Olympiad I-1

9/20/2014
Determine all functions f:R→Rf:\mathbb{R} \to \mathbb{R} such that xf(y)+f(xf(y))−xf(f(y))−f(xy)=2x+f(y)−f(x+y) xf(y) + f(xf(y)) - xf(f(y)) - f(xy) = 2x + f(y) - f(x+y) holds for all x,y∈Rx,y \in \mathbb{R}.
functionalgebra proposedalgebra
lowest possible value of the expression

Source: Middle European Mathematical Olympiad T-1

9/21/2014
Determine the lowest possible value of the expression 1a+x+1a+y+1b+x+1b+y \frac{1}{a+x} + \frac{1}{a+y} + \frac{1}{b+x} + \frac{1}{b+y} where a,b,x,a,b,x, and yy are positive real numbers satisfying the inequalities 1a+x≥12 \frac{1}{a+x} \ge \frac{1}{2} 1a+y≥12\frac{1}{a+y} \ge \frac{1}{2} 1b+x≥12 \frac{1}{b+x} \ge \frac{1}{2} 1b+y≥1. \frac{1}{b+y} \ge 1.
inequalitiesinequalities proposed