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Iran TST

Source: Iranian TST 2019, first exam day 2, problem 6

June 24, 2019
number theory

Problem Statement

{an}n≥0\{a_{n}\}_{n\geq 0} and {bn}n≥0\{b_{n}\}_{n\geq 0} are two sequences of positive integers that ai,bi∈{0,1,2,⋯ ,9}a_{i},b_{i}\in \{0,1,2,\cdots,9\}. There is an integer number MM such that an,bn≠0a_{n},b_{n}\neq 0 for all n≥Mn\geq M and for each n≥0n\geq 0 (an⋯a1a0‾)2+999∣(bn⋯b1b0‾)2+999(\overline{a_{n}\cdots a_{1}a_{0}})^{2}+999 \mid(\overline{b_{n}\cdots b_{1}b_{0}})^{2}+999 prove that an=bna_{n}=b_{n} for n≥0n\geq 0.\\ (Note that (xnxn−1…x0‾)=10n×xn+⋯+10×x1+x0(\overline{x_nx_{n-1}\dots x_0}) = 10^n\times x_n + \dots + 10\times x_1 + x_0.)
Proposed by Yahya Motevassel
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