We solve problems

Problems Methods Pigeonhole principle Pigeonhole principle (area and volume)

Leo’s grandma placed five empty plates on a square 1 metre $\times$ 1 metre table for dinner. Show that some two of these plates were less than 75 cm apart.

Problem #PRU-58107

Problems Geometry Plane geometry Area Finding the area by rearrangment Set theory and logic Set theory Inclusion-exclusion principle Methods Pigeonhole principle Pigeonhole principle (area and volume) Proof by contradiction

14–15 3.0

a) A square of area 6 contains three polygons, each of area 3. Prove that among them there are two polygons that have an overlap of area no less than 1.

b) A square of area 5 contains nine polygons of area 1. Prove that among them there are two polygons that have an overlap of area no less than $\frac{1}{9}$ .

Problem #PRU-65155

Problems Methods Pigeonhole principle Pigeonhole principle (area and volume)

14–16 3.0

A carpet has a square shape with side 275 cm. A moth has eaten 4 holes through it. Will it always be possible to cut a square section of side 1 m out of the carpet, so that the section does not contain any holes? Treat the holes as points.

Problem #PRU-65194

Problems Combinatorics Dissections, partitions, covers and tilings Dissections with certain properties Methods Pigeonhole principle Pigeonhole principle (area and volume)

14–15 4.0

Every day, James bakes a square cake size $3 \times 3$ . Jack immediately cuts out for himself four square pieces of size $1 \times 1$ with sides parallel to the sides of the cake (not necessarily along the $3 \times 3$ grid lines). After that, Sarah cuts out from the rest of the cake a square piece with sides, also parallel to the sides of the cake. What is the largest piece of cake that Sarah can count on, regardless of Jack’s actions?

Problem #PRU-65204

Problems Geometry Plane geometry Triangles Similar triangles Auxiliary similar 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.

Problem #PRU-108995

Problems Combinatorics Packing problems Methods Pigeonhole principle Pigeonhole principle (area and volume) Systems of points and line segments Systems of points

13–15 3.0

Prove that, in a circle of radius 10, you cannot place 400 points so that the distance between each two points is greater than 1.

Problem #PRU-32068

Problems Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (area and volume)

11–13 3.0

A square area of size $100 \times 100$ is covered in tiles of size $1 \times 1$ in 4 different colours – white, red, black, and grey. No two tiles of the same colour touch one another, that is share a side or a corner. How many red tiles can there be?

Problem #PRU-32075

Problems Combinatorics Geometry on grid paper Methods Pigeonhole principle Pigeonhole principle (area and volume) Dissections, partitions, covers and tilings Various dissection problems

12–14 4.0

One corner square was cut from a chessboard. What is the smallest number of equal triangles that can be cut into this shape?