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Iran MO Number theory

Source: Iran second round 2018 day 1 problem 3

April 26, 2018
number theory

Problem Statement

Let a>ka>k be natural numbers and r1<r2<…rn,s1<s2<⋯<snr_1<r_2<\dots r_n,s_1<s_2<\dots <s_n be sequences of natural numbers such that:
(ar1+k)(ar2+k)…(arn+k)=(as1+k)(as2+k)…(asn+k)(a^{r_1}+k)(a^{r_2}+k)\dots (a^{r_n}+k)=(a^{s_1}+k)(a^{s_2}+k)\dots (a^{s_n}+k)
Prove that these sequences are equal.
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