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Numbers from 11 to 99 are written on 99 cards

Source:

September 22, 2010
combinatoricsmodular arithmeticIMO Shortlist

Problem Statement

Natural numbers from 11 to 9999 (not necessarily distinct) are written on 9999 cards. It is given that the sum of the numbers on any subset of cards (including the set of all cards) is not divisible by 100100. Show that all the cards contain the same number.
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