TheColoringMethod

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2470   
Colour all the points on a plane either white or black randomly. Show that it is always possible to find a triangle whose vertices have the same colour and its side length is either $1$ or $\sqrt{3}$.

2471   
Randomly colour all the points one a plane either black or white. Show that if any two points with a distance of $2$ units have the same colour, then all the points on this plane have the same colour.

2744   

Joe cuts the top left corner and the bottom right corner of an $8\times 8$ board, and then try to cover the remaining board using thirty-one $1\times 2$ smaller pieces. Is it possible? Note: a smaller piece can be rotated, but cannot be further broken up.



2745   

Joe cuts a $2\times 2$ corner from an $8\times 8$ board, and then try to cover the remaining part using $15$ smaller L-shaped grids made of $4$ grids. Is it possible?



2747   
Show that among any $6$ people in the world, there must exist $3$ people who either know each other or do not know each other.

2748   
There are $6$ points in the $3$-D space. No three points are on the same line and no four points are one the same plane. Hence totally $15$ segments can be created among these points. Show that if each of these $15$ segments is colored either black or white, there must exist a triangle whose sides are of same color.

2749   
Seventeen people correspond by mail with one another - each one with all the rest. In their letters only three different topics are discussed. Each pair of correspondents deals with only one of these topics. Prove that there are at least three people who write to each other about the same topic.

2759   
Show that it is impossible to cover an $8\times 8$ square using fifteen $4\times 1$ rectangles and one $2\times 2$ square.

2760   
Is it possible to arrange these numbers, $1, 1, 2, 2, 3, 3, \cdots, 1986, 1986$ to form a sequence for such there is $1$ number between two $1$'s, $2$ numbers between two $2$'s, $\cdots$, $1986$ numbers between two 1986's?

2763   
It is possible to cover a $6\times 6$ grid using one L-shaped piece made of 3 grids and eleven $3\times 1$ smaller grid ?

3929   
Is it possible to use twenty seven $1\times 2\times 4$ blocks to construct a $6\times 6\times 6$ cube?

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