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Can you color every positive integer either black or white such that there are no entirely white or entirely black non-constant infinite arithmetic progressions?
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The red, blue, and green circles have diameters 3, 4, and 5, respectively.
What is the radius of the black circle tangent to all three of these circles (to 5 decimal places)?
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If you took a solid, 4-dimensional hypercube and cut it through with a single, flat, 3D hyperplane, which could not be the resulting cross-section?
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You have to fill a large bag with 2018 fruits—apples, bananas, cantaloupes, durians, and raspberries—under the following restrictions:
How many different ways are there to fill the bag?
Note: This problem can be solved without having to grind all the possibilities. Generating functions may help you.
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You are in room $1$ in the upper left corner of a maze of rooms. The goal is to escape this maze and exit through room $25$ at the bottom right, but you can only move either right or down, passing through a total of $9$ rooms.
You need to deliberately find a path to get out exactly the same way you were when you started in room $1,$ because each room alters you, sometimes even making you imaginary.
What path will let you do that?
Enter your answer as the total of the room numbers you must pass through in order to do that, a sum of $9$ numbers starting with $1$ and ending with $25$.
For example, if you go through rooms $1,2,3,4,5,10,15,20,25$, the total is $85$. However, you will experience the succession of functions
$\text{Tan}(\text{Sin}(\text{ArcSech}(\text{ArcCsc}(\text{Coth}(\text{ArcCosh}(\text{Sec}(\text{ArcCsch}(x)))))))))$
so that if you started with for example a real value of $0.5,$ you would end up being a totally imaginary number $0.714i$ when you leave the maze, which does not fulfill the requirement.
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