2

Part of 2004 Rioplatense Mathematical Olympiad, Level 3

Problems(2)

Subsets of {1,...,2004} with a & b such that 2004 | a^2-b^2

Source: XIII Rioplatense Mathematical Olympiad (2004), Level 3

Find the smallest integer

n

such that each subset of

\{1,2,\ldots, 2004\}

n

elements has two distinct elements

a

b

a^2-b^2

is a multiple of

2004

modular arithmeticnumber theory unsolvednumber theory

Arranging cardboard circles on a table

Source: XIII Rioplatense Mathematical Olympiad (2004), Level 3

A collection of cardboard circles, each with a diameter of at most

1

5\times 8

table without overlapping or overhanging the edge of the table. A cardboard circle of diameter

2

is added to the collection. Prove that this new collection of cardboard circles can be placed on a

7\times 7

table without overlapping or overhanging the edge.

geometryrectanglecombinatorics unsolvedcombinatorics