2006 Junior Balkan Team Selection Tests - Romania

Part of JBMO TST - Romania

Subcontests

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The set of positive integers is partitioned in infinite subs

The set of positive integers is partitionated in subsets with infinite elements each. The question (in each of the following cases) is if there exists a subset in the partition such that any positive integer has a multiple in this subset. a) Prove that if the number of subsets in the partition is finite the answer is yes. b) Prove that if the number of subsets in the partition is infinite, then the answer can be no (for a certain partition).

Prove that the set of real numbers can be partitioned

Prove that the set of real numbers can be partitioned in (disjoint) sets of two elements each.

4a^2 + r(b) and 4b^2 + r(a) can not be simultaneously squares

For a positive integer

n

r(n)

the number having the digits of

n

in reverse order- for example,

r(2006) = 6002

. Prove that for any positive integers a and b the numbers

4a^2 + r(b)

4b^2 + r(a)

can not be simultaneously squares.

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