Logic
# Arithmetic Puzzles

Let \(N\) denote the concatenation of the first 60 positive integers:

\[ N = 1234567891011121314...585960. \]

Remove any 100 digits from \(N\) without rearranging the remaining digits, and call the resulting number \(M\). What is the largest possible value of \(M\)?

You are the village chief. Your village has 1000 residents including yourself. One day the ferocious EVEN DEMON attacks your village and snarls,

"All you 1000 villagers stand in a large CIRCLE, numbering yourself from 1 to 1000. Starting from number 1, I will eat every second villager (which means he will eat villager number 2, then villager number 4 and so on) and keep going around and around the circle eating up every second villager and keep doing this till there is only 1 villager left. That last villager I will spare and he is free to escape."

Since you are the village chief, you have the right to choose where you wish to stand.

In the original circle of 1000 villagers which number will you choose to stand at to be the last villager standing and escape the clutches of the EVEN DEMON?

This problem is not an original. It is adapted from a famous problem recorded by a Jewish historian.

Five points are located on a line. The ten distances between the pairs of points are listed from smallest to largest:

\[\large{2, \ 4, \ 5,\ 7,\ 8,\ K,\ 13,\ 15,\ 17,\ 19}\]

Determine the value of \(K\).

**integer** value of the expression?

Last night I went with my wife to a party where four other married couples were present.

Every person shook hands with only the people he or she was NOT acquainted with. (Obviously, no one shook his or her own hand or spouse's hand, and no one shook hands with the same person twice.)

When the handshaking was over, I asked everyone including my wife how many hands they each shook.

To my surprise, I got 9 different answers!

How many hands did my wife shake?

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