MathDB

2

Part of 2018 Bundeswettbewerb Mathematik

Problems(2)

An equation involving the floor function

Source: Bundeswettbewerb Mathematik 2018, Round 1 - #2

6/30/2018
Find all real numbers xx satisfying the equation ⌊20x+18⌋+⌊x+1820⌋=1.\left\lfloor \frac{20}{x+18}\right\rfloor+\left\lfloor \frac{x+18}{20}\right\rfloor=1.
algebraequationreal numbersfloor functionfunction
Possible values for a sum under some functional equation

Source: Bundeswettbewerb Mathematik 2018, Round 2, Problem 2

2/5/2019
Consider all functions f:R→Rf:\mathbb{R} \to \mathbb{R} satisfying f(1−f(x))=xf(1-f(x))=x for all x∈Rx \in \mathbb{R}. a) By giving a concrete example, show that such a function exists. b) For each such function define the sum Sf=f(−2017)+f(−2016)+⋯+f(−1)+f(0)+f(1)+⋯+f(2017)+f(2018).S_f=f(-2017)+f(-2016)+\dots+f(-1)+f(0)+f(1)+\dots+f(2017)+f(2018). Determine all possible values of SfS_f.
functionalgebraalgebra proposedfunctional equationSum