MathDB

3

Part of 2005 International Zhautykov Olympiad

Problems(4)

partitioning 2n points

Source: IZO 1 Junior Problem 3

12/16/2008
Let A A be a set of 2n 2n points on the plane such that no three points are collinear. Prove that for any distinct two points a,bāˆˆA a,b\in A there exists a line that partitions A A into two subsets each containing n n points and such that a,b a,b lie on different sides of the line.
rotationcombinatorics proposedcombinatorics
Primes

Source: IZO 1 Junior Problem 6

12/16/2008
Find all prime numbers p,q p,q less than 2005 and such that q|p^2 \plus{} 4, p|q^2 \plus{} 4.
number theoryrelatively primenumber theory unsolved
Find set of all points for a regular triangular pyramid

Source: First Zhautykov Olympiad 2005, Problem 3

12/22/2008
Let SABC be a regular triangular pyramid. Find the set of all points D (D! \equal{} S) in the space satisfing the equation |cos ASD \minus{} 2cosBSD \minus{} 2 cos CSD| \equal{} 3.
geometry3D geometrypyramidgeometry unsolved
Prime numbers less than 2005 with q | p^2 + 8

Source: First Zhautykov Olympiad 2005, Problem 6

12/22/2008
Find all prime numbers p,q<2005 p,q < 2005 such that q | p^{2} \plus{} 8 and p|q^{2} \plus{} 8.
Vietanumber theory unsolvednumber theory