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Magician asked a spectator to think of a three-digit number

Source: Tuymaada 2009, Senior League, Second Day, Problem 1

July 19, 2009
modular arithmeticalgebra unsolvedalgebra

Problem Statement

A magician asked a spectator to think of a three-digit number abc‾ \overline{abc} and then to tell him the sum of numbers acb‾ \overline{acb}, bac‾ \overline{bac}, bca‾ \overline{bca}, cab‾ \overline{cab}, and cba‾ \overline{cba}. He claims that when he knows this sum he can determine the original number. Is that so?
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