IMO Shortlist 1971 problem 15
Dodao/la:
arhiva2. travnja 2012. Natural numbers from
![1](/media/m/a/9/1/a913f49384c0227c8ea296a725bfc987.png)
to
![99](/media/m/1/e/7/1e7b28b7e4fcbc47967bc9a712d7edc6.png)
(not necessarily distinct) are written on
![99](/media/m/1/e/7/1e7b28b7e4fcbc47967bc9a712d7edc6.png)
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
![100](/media/m/c/c/c/ccc0563efabf7c1a3d81b0dc63f5b627.png)
. Show that all the cards contain the same number.
%V0
Natural numbers from $1$ to $99$ (not necessarily distinct) are written on $99$ 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 $100$. Show that all the cards contain the same number.
Izvor: Međunarodna matematička olimpijada, shortlist 1971