Olympiad problems

2011 Indonesia TST, 1

Let $Q^+$ denote the set of positive rationals. Determine all functions $f : Q^+ \to Q^+$ that satisfy both of these conditions: (i) $f(x)$ is an integer if and only if $x$ is an integer; (ii) $f(f(xf(y)) + x) = yf(x) + x$ for all $x, y \in Q^+$.

2012 IMAC Arhimede, 3

Find all functions $f:Q^+ \to Q^+$ such that for any $x,y \in Q^+$ : $$y=\frac{1}{2}\left[f\left(x+\frac{y}{x}\right)- \left(f(x)+\frac{f(y)}{f(x)}\right)\right]$$

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2010 Belarus Team Selection Test, 2.4

Find all functions $f, g : Q \to Q$ satisfying the following equality $f(x + g(y)) = g(x) + 2 y + f(y)$ for all $x, y \in Q$. (I. Voronovich)

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2011 Indonesia TST, 1

Let $Q^+$ denote the set of positive rationals. Determine all functions $f : Q^+ \to Q^+$ that satisfy both of these conditions: (i) $f(x)$ is an integer if and only if $x$ is an integer; (ii) $f(f(xf(y)) + x) = yf(x) + x$ for all $x, y \in Q^+$.

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1991 Irish Math Olympiad, 4

Let $\mathbb{P}$ be the set of positive rational numbers and let $f:\mathbb{P}\to\mathbb{P}$ be such that $$f(x)+f\left(\frac{1}{x}\right)=1$$ and $$f(2x)=2f(f(x))$$ for all $x\in\mathbb{P}$. Find, with proof, an explicit expression for $f(x)$ for all $x\in \mathbb{P}$.

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