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Does Not Divide Another

How many numbers between 1 and 100 can you pick at most, so that none of them divide another?

Murdered Wife

One night, a man received a call from the Police. The Police told the man that his wife was murdered, and that he should get to the crime scene as soon as possible. The man immediately hung up the phone and drove his car for 20 minutes. As soon as he got to the crime scene, the Police arrested him, and he got convicted for murder.

How did the Police know that the man committed the crime?

Self-Describing Number

Find all 10-digit numbers with the following property:

  • the first digit shows the number of 0s in the number
  • the second digit shows the number of 1s in the number
  • the third digit shows the number of 2s in the number, and so on
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Five Points in a Square

There are 5 points in a square 1×1. Show that 2 of the points are within distance 0.75.