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Found: 13

Problems Geometry Plane geometry Triangles Similar triangles Auxiliary simialr triangles Methods Extremal principle Extremal principle (other) Pigeonhole principle Pigeonhole principle (area and volume)

14–16 4.0

A unit square is divided into \(n\) triangles. Prove that one of the triangles can be used to completely cover a square with side length \(\frac{1}{n}\).

Problem #PRU-78270

Problems Geometry Plane geometry Area Area of curvilinear shapes Methods Pigeonhole principle Pigeonhole principle (area and volume)

13–15 4.0

120 unit squares are placed inside a \(20 \times 25\) rectangle. Prove that it will always be possible to place a circle with diameter 1 inside the rectangle, without it overlapping with any of the unit squares.

Problem #PRU-79322

Problems Geometry Solid geometry Spheres Circle on a sphere Methods Pigeonhole principle Pigeonhole principle (area and volume) Extremal principle Smallest or largest distance (length)

14–16 4.0

A spherical sun is observed to have a finite number of circular sunspots, each of which covers less than half of the sun’s surface. These sunspots are said to be enclosed, that is no two sunspots can touch, and they do not overlap with one another. Prove that the sun will have two diametrically opposite points that are not covered by sunspots.