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non-negative subadditive FE over the rationals

Source: 2020 Kürschák Competition P2

April 9, 2021
functional equationalgebra

Problem Statement

Find all functions f ⁣:Q→R≥0f\colon \mathbb{Q}\to \mathbb{R}_{\geq 0} such that for any two rational numbers xx and yy the following conditions hold
[*] f(x+y)≤f(x)+f(y)f(x+y)\leq f(x)+f(y), [*]f(xy)=f(x)f(y)f(xy)=f(x)f(y), [*]f(2)=1/2f(2)=1/2.
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