MathDB

1

Part of 2009 Pan African

Problems(2)

Numbers in the set {-1,1} such that sum of pairs is 999

Source: Pan African Olympiad 2009

10/1/2011
Determine whether or not there exist numbers x1,x2,…,x2009x_1,x_2,\ldots ,x_{2009} from the set {−1,1}\{-1,1\}, such that: x1x2+x2x3+x3x4+…+x2008x2009+x2009x1=999x_1x_2+x_2x_3+x_3x_4+\ldots+x_{2008}x_{2009}+x_{2009}x_1=999
number theory proposednumber theory
Minimum number of colours used for hats and ribbons

Source: Pan African Olympiad 2009

10/1/2011
Consider nn children in a playground, where n≥2n\ge 2. Every child has a coloured hat, and every pair of children is joined by a coloured ribbon. For every child, the colour of each ribbon held is different, and also different from the colour of that child’s hat. What is the minimum number of colours that needs to be used?
combinatorics proposedcombinatorics