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Romania TST 2021 Day 3 P3

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June 14, 2021
algebraSequenceromaniaRomanian TSTTST

Problem Statement

Let α\alpha be a real number in the interval (0,1).(0,1). Prove that there exists a sequence (εn)n≥1(\varepsilon_n)_{n\geq 1} where each term is either 00 or 11 such that the sequence (sn)n≥1(s_n)_{n\geq 1} sn=ε1n(n+1)+ε2(n+1)(n+2)+...+εn(2n−1)2ns_n=\frac{\varepsilon_1}{n(n+1)}+\frac{\varepsilon_2}{(n+1)(n+2)}+...+\frac{\varepsilon_n}{(2n-1)2n}verifies the inequality 0≤α−2nsn≤2n+10\leq \alpha-2ns_n\leq\frac{2}{n+1} for any n≥2.n\geq 2.
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