This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 2

2014 Federal Competition For Advanced Students, 2
Tags: combinatorics , lattice points , square lattice , lattice
We call a set of squares with sides parallel to the coordinate axes and vertices with integer coordinates friendly if any two of them have exactly two points in common. We consider friendly sets in which each of the squares has sides of length $n$. Determine the largest possible number of squares in such a friendly set.

1997 ITAMO, 3
Tags: combinatorial geometry , combinatorics , lattice , square lattice , coloring
The positive quadrant of a coordinate plane is divided into unit squares by lattice lines. Is it possible to color the squares in black and white so that: (i) In every square of side $n$ ($n \in N$) with a vertex at the origin and sides are parallel to the axes, there are more black than white squares; (ii) Every diagonal parallel to the line $y = x$ intersects only finitely many black squares?