grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the 
cars from the 
be a convex hexagon, such that the triangles 
and 
are equilateral and the diagonals 
and 
are concurrent. Prove that 
or 
is prime.
Y by Adventure10, Mango247, and 1 other user
An 
board is coloured in 
colours such that the main diagonal (from top-left to bottom-right) is coloured in the first colour; the two adjacent diagonals are coloured in the second colour; the two next diagonals (one from above and one from below) are coloured in the third colour, etc; the two corners (top-right and bottom-left) are coloured in the -th colour. It happens that it is possible to place on the board rooks, no two attacking each other and such that no two rooks stand on cells of the same colour. Prove that 
or 
.
board is coloured in 
colours such that the main diagonal (from top-left to bottom-right) is coloured in the first colour; the two adjacent diagonals are coloured in the second colour; the two next diagonals (one from above and one from below) are coloured in the third colour, etc; the two corners (top-right and bottom-left) are coloured in the -th colour. It happens that it is possible to place on the board rooks, no two attacking each other and such that no two rooks stand on cells of the same colour. Prove that 
or 
.
be the position of the rook in the 
t
is a permutation of 
.
must all be different and take each value from 
once. So we have![\[\binom n2=\sum_{i=1}^n|x_i-i|\equiv\sum_{i=1}^nx_i+\sum_{i=1}^ni=2\sum_{i=1}^ni\equiv0\pmod 2.\]](http://latex.artofproblemsolving.com/1/1/9/11925641257809cf6c78a11e19505e25268245b5.png)
So 
qed. 
