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Part of 2010 Argentina Team Selection Test

Problems(1)

Divide {1,2,...,2010} in subsets - Argentina TST 2010

Source:

5/3/2010
Suppose a1,a2,...,ara_1, a_2, ..., a_r are integers with ai≥2a_i \geq 2 for all ii such that a1+a2+...+ar=2010a_1 + a_2 + ... + a_r = 2010. Prove that the set {1,2,3,...,2010}\{1,2,3,...,2010\} can be partitioned in rr subsets A1,A2,...,ArA_1, A_2, ..., A_r each with a1,a2,...,ara_1, a_2, ..., a_r elements respectively, such that the sum of the numbers on each subset is divisible by 20112011. Decide whether this property still holds if we replace 20102010 by 20112011 and 20112011 by 20122012 (that is, if the set to be partitioned is {1,2,3,...,2011}\{1,2,3,...,2011\}).
algebrapolynomialinductioncombinatorics unsolvedcombinatorics