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Spain National Mathematical Olympiad 2016 Day 1 Problem 1

Source: Spain Math Olympiad

April 2, 2016
algebranational olympiadOlympiadarithmetic sequencegeometric sequence

Problem Statement

Two real number sequences are guiven, one arithmetic (an)n∈N\left(a_n\right)_{n\in \mathbb {N}} and another geometric sequence (gn)n∈N\left(g_n\right)_{n\in \mathbb {N}} none of them constant. Those sequences verifies a1=g1≠0a_1=g_1\neq 0, a2=g2a_2=g_2 and a10=g3a_{10}=g_3. Find with proof that, for every positive integer pp, there is a positive integer mm, such that gp=amg_p=a_m.
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