2

Part of 2002 Moldova Team Selection Test

Problems(3)

There exists a partition of the set

Source: Moldova TST 2002 - E1 - P2

Prove that there exists a partition of the set

A = \{1^3, 2^3, \ldots , 2000^3\}

19

nonempty subsets such that the sum of elements of each subset is divisible by

2001^2

combinatorics unsolvedcombinatorics

Source: Moldova IMO TST 2002

A

be a set containing

4k

consecutive positive integers, where

k \geq 1

is an integer. Find the smallest

k

for which the set A can be partitioned into two subsets having the same number of elements, the same sum of elements, the same sum of the squares of elements, and the same sum of the cubes of elements.

Moldova Team Selection Test

Source: Moldova IMO TST 2002

S= \{ a_1, \ldots, a_n\}

n\geq 1

positive real numbers. For each nonempty subset of

S

the sum of its elements is written down. Show that all written numbers can be divided into

n

classes such that in each class the ratio of the greatest number to the smallest number is not greater than

2