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Found: 14

Problems Combinatorics Geometry on grid paper Integer lattices Integer lattices (other) Painting problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 3.0

All of the points with whole number co-ordinates in a plane are plotted in one of three colours; all three colours are present. Prove that there will always be possible to form a right-angle triangle from these points so that its vertices are of three different colours.

Problem #PRU-111000

Problems Combinatorics Integer lattices Integer lattices (other) Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 3.0

Inside a square with side 1 there are several circles, the sum of the radii of which is 0.51. Prove that there is a line that is parallel to one side of the square and that intersects at least 2 circles.

Problem #PRU-30748

Problems Combinatorics Integer lattices Polygons and polyhedra with vertices in lattice points

12–13 2.0

Find the number of rectangles made up of the cells of a board with \(m\) horizontals and \(n\) verticals that contain a cell with the coordinates \((p, q)\).

Problem #PRU-35753

Problems Combinatorics Integer lattices Integer lattices (other) Methods Pigeonhole principle Pigeonhole principle (other)

13–15 3.0

A convex polygon on a plane contains no fewer than \(m^2+1\) points with whole number co-ordinates. Prove that within the polygon there are \(m+1\) points with whole number co-ordinates that lie on a single straight line.