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MMO 269 Moscow MO 1954 system inequalities 100x100

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August 13, 2019
inequalitiessystem of inequalitiesalgebra

Problem Statement

a) Given 100100 numbers a1,...,a100a_1, ..., a_{100} such that {a1−3a2+2a3≥0,a2−3a3+2a4≥0,a3−3a4+2a5≥0,...a99−3a100+2a1≥0,a100−3a1+2a2≥0\begin{cases} a_1 - 3a_2 + 2a_3 \ge 0, \\ a_2 - 3a_3 + 2a_4 \ge 0, \\ a_3 - 3a_4 + 2a_5 \ge 0, \\ ... \\ a_{99} - 3a_{100} + 2a_1 \ge 0, \\ a_{100} - 3a_1 + 2a_2 \ge 0 \end{cases} prove that the numbers are equal.
b) Given numbers a1=1,...,a100a_1=1, ..., a_{100} such that {a1−4a2+3a3≥0,a2−4a3+3a4≥0,a3−4a4+3a5≥0,...a99−4a100+3a1≥0,a100−4a1+3a2≥0\begin{cases} a_1 - 4a_2 + 3a_3 \ge 0, \\ a_2 - 4a_3 + 3a_4 \ge 0, \\ a_3 - 4a_4 + 3a_5 \ge 0, \\ ... \\ a_{99} - 4a_{100} + 3a_1 \ge 0, \\ a_{100} - 4a_1 + 3a_2 \ge 0 \end{cases} Find a2,a3,...,a100.a_2, a_3, ... , a_{100}.
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