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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.

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.

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 $\frac{1}{n}$.

Every day, James bakes a square cake size $3\times3$. Jack immediately cuts out for himself four square pieces of size $1\times1$ with sides parallel to the sides of the cake $($not necessarily along the $3\times3$ 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?

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

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 $\frac{1}{9}$.

A square of side 15 contains 20 non-overlapping unit squares. Prove that it is possible to place a circle of radius 1 inside the large square, so that it does not overlap with any of the unit squares.

On the planet Tau Ceti, the landmass takes up more than half the surface area. Prove that the Tau Cetians can drill a hole through the centre of their planet that connects land to land.

In a square which has sides of length 1 there are 100 figures, the total area of which sums to more than 99. Prove that in the square there is a point which belongs to all of these figures.

10 magazines lie on a coffee table, completely covering it. Prove that you can remove five of them so that the remaining magazines will cover at least half of the table.

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

A square area of size 100×100 is covered in tiles of size 1×1 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?

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.