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Sets of solutions to floor equation

Source: Romanian MO 2010 Grade 9

August 6, 2012
floor functionalgebra unsolvedalgebra

Problem Statement

For any integer n≥2n\ge 2 denote by AnA_n the set of solutions of the equation x=⌊x2⌋+⌊x3⌋+⋯+⌊xn⌋.x=\left\lfloor\frac{x}{2}\right\rfloor+\left\lfloor\frac{x}{3}\right\rfloor+\cdots+\left\lfloor\frac{x}{n}\right\rfloor . a) Determine the set A2∪A3A_2\cup A_3. b) Prove that the set A=⋃n≥2AnA=\bigcup_{n\ge 2}A_n is finite and find max⁡A\max A.
Dan Nedeianu & Mihai Baluna
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