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Romanian National Olympiad 2018 - Grade 10 - problem 4

Source: Romania NMO - 2018

April 12, 2018
algebra

Problem Statement

Let n∈N≥2.n \in \mathbb{N}_{\geq 2}. For any real numbers a1,a2,...,ana_1,a_2,...,a_n denote S0=1S_0=1 and for 1≤k≤n1 \leq k \leq n denote Sk=∑1≤i1<i2<...<ik≤nai1ai2...aikS_k=\sum_{1 \leq i_1 < i_2 < ... <i_k \leq n}a_{i_1}a_{i_2}...a_{i_k} Find the number of n−n-tuples (a1,a2,...an)(a_1,a_2,...a_n) such that (Sn−Sn−2+Sn−4−...)2+(Sn−1−Sn−3+Sn−5−...)2=2nSn.(S_n-S_{n-2}+S_{n-4}-...)^2+(S_{n-1}-S_{n-3}+S_{n-5}-...)^2=2^nS_n.
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