1

Part of 2008 Rioplatense Mathematical Olympiad, Level 3

Problems(2)

0-1 matrix w/ rows & cols having the same number of 0s or 1s

Source: XVII Olimpíada Matemática Rioplatense (2008)

In each square of a chessboard with

a

b

0

1

is written satisfying the following conditions. [*]If a row and a column intersect in a square with a

0

, then that row and column have the same number of

0

s. [*]If a row and a column intersect in a square with a

1

, then that row and column have the same number of

1

s. Find all pairs

(a,b)

for which this is possible.

linear algebramatrixcombinatorics unsolvedcombinatorics

Partition N so that k, 2k,..., 12k are in different subsets

Source: XVII Olimpíada Matemática Rioplatense (2008)

Can the positive integers be partitioned into

12

subsets such that for each positive integer

k

k, 2k,\ldots,12k

belong to different subsets?

modular arithmeticparameterizationnumber theorynumber theory unsolved