olymmath

ISL 2023 A4

  • Answer

Let R>0\mathbb{R}_{>0} be the set of positive real numbers. Determine all functions f:R>0→R>0f: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0} such that x(f(x)+f(y))⩾(f(f(x))+y)f(y)x(f(x)+f(y)) \geqslant(f(f(x))+y) f(y) for every x,y∈R>0x, y \in \mathbb{R}_{>0}. (Belgium)

Dataset text NuminaMath olympiads. Quoted with source.

Official text: IMO 2023 Shortlist booklet, page 5.

Show the answer

f(x) = \frac{c}{x}

Solutions

Official solution Solution 1 by the IMO Problem Selection Committee

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