MathDB

3

Part of 2012 India IMO Training Camp

Problems(3)

Prove f(x+y+xy)=f(x)+f(y)+f(xy) implies f(x+y)=f(x)+f(y)

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7/17/2012
Let f:R⟶Rf:\mathbb{R}\longrightarrow \mathbb{R} be a function such that f(x+y+xy)=f(x)+f(y)+f(xy)f(x+y+xy)=f(x)+f(y)+f(xy) for all x,y∈Rx, y\in\mathbb{R}. Prove that ff satisfies f(x+y)=f(x)+f(y)f(x+y)=f(x)+f(y) for all x,y∈Rx, y\in\mathbb{R}.
functionalgebra proposedalgebra
Number of 6-tuples satisfying conditions

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7/17/2012
How many 66-tuples (a,b,c,d,e,f)(a, b, c, d, e, f) of natural numbers are there for which a>b>c>d>e>fa>b>c>d>e>f and a+f=b+e=c+d=30a+f=b+e=c+d=30 are simultaneously true?
Determine function given two inequalities

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7/23/2012
Let R+\mathbb{R}^{+} denote the set of all positive real numbers. Find all functions f:R+⟶Rf:\mathbb{R}^{+}\longrightarrow \mathbb{R} satisfying f(x)+f(y)≤f(x+y)2,f(x)x+f(y)y≥f(x+y)x+y,f(x)+f(y)\le \frac{f(x+y)}{2}, \frac{f(x)}{x}+\frac{f(y)}{y}\ge \frac{f(x+y)}{x+y}, for all x,y∈R+x, y\in \mathbb{R}^{+}.
functioninequalitiesinductionalgebra proposedalgebra