MathDB

3

Part of 1990 Polish MO Finals

Problems(2)

Tournament and diving into groups

Source: Problem 3, Polish NO 1990

9/30/2005
In a tournament, every two of the nn players played exactly one match with each other (no draws). Prove that it is possible either (i) to partition the league in two groups AA and BB such that everybody in AA defeated everybody in BB; or (ii) to arrange all the players in a chain x1,x2,...,xn,x1x_1, x_2, . . . , x_n, x_1 in such a way that each player defeated his successor.
graph theorycombinatorics unsolvedcombinatorics
Sum divisible by 3

Source: Problem 6, Polish NO 1990

9/30/2005
Prove that for all integers n>2n > 2, 3∣∑i=0[n/3](−1)iCn3i 3| \sum\limits_{i=0}^{[n/3]} (-1)^i C _n ^{3i}
modular arithmeticnumber theory unsolvednumber theory