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Romanian District Olympiad 2019 - Grade 9 - Problem 3

Source: Romanian District Olympiad 2019 - Grade 9 - Problem 3

March 18, 2019
Sequencefloor functionArithmetic Progressionalgebra

Problem Statement

Let (an)n∈N(a_n)_{n \in \mathbb{N}} be a sequence of real numbers such that 2(a1+a2+…+an)=nan+1 ∀ n≥1.2(a_1+a_2+…+a_n)=na_{n+1}~\forall~n \ge 1. <spanclass=′latex−bold′>a)</span><span class='latex-bold'>a)</span> Prove that the given sequence is an arithmetic progression. <spanclass=′latex−bold′>b)</span><span class='latex-bold'>b)</span> If ⌊a1⌋+⌊a2⌋+…+⌊an⌋=⌊a1+a2+…+an⌋ ∀ n∈N,\lfloor a_1 \rfloor + \lfloor a_2 \rfloor +…+ \lfloor a_n \rfloor = \lfloor a_1+a_2+…+a_n \rfloor~\forall~ n \in \mathbb{N}, prove that every term of the sequence is an integer.
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