MathDB

5

Part of 2008 All-Russian Olympiad

Problems(3)

Cells of a chessboard with distance 50

Source: All-Russian Olympiad 2008 9.5

6/14/2008
The distance between two cells of an infinite chessboard is defined as the minimum nuber to moves needed for a king for move from one to the other.One the board are chosen three cells on a pairwise distances equal to 100 100. How many cells are there that are on the distance 50 50 from each of the three cells?
analytic geometrycombinatorics proposedcombinatorics
Find all triplets satisfying to inequalities.

Source: All-Russian Olympiad 2008 10.5

6/14/2008
Determine all triplets of real numbers x,y,z x,y,z satisfying 1\plus{}x^4\leq 2(y\minus{}z)^2,  1\plus{}y^4\leq 2(x\minus{}z)^2,  1\plus{}z^4\leq 2(x\minus{}y)^2.
algebrasystem of equationsalgebra proposed
integral roots

Source: All Russian Mathematical Olympiad 2008. 11.5

6/13/2008
The numbers from 51 51 to 150 150 are arranged in a 10×10 10\times 10 array. Can this be done in such a way that, for any two horizontally or vertically adjacent numbers a a and b b, at least one of the equations x^2 \minus{} ax \plus{} b \equal{} 0 and x^2 \minus{} bx \plus{} a \equal{} 0 has two integral roots?
calculusintegrationinequalitiesnumber theory proposednumber theory