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2018-09-17 Intermediate

         

A number of the form \(10^n-1=\underbrace{9999...9}_{n\text{ times}},\) where \(n\) is a positive integer, will never be divisible by \(2\) or \(5.\)

Are there any other prime numbers that numbers of this form are never divisible by?

A circle is inscribed in a right trapezoid with base lengths 22 and 35.

What is the area of this trapezoid?

\[\LARGE \begin{align}\large \sqrt{2^{\sqrt{2^{\sqrt{2^{\sqrt {2^{\sqrt 2}}}}}}}} = \sqrt 2^{\sqrt 2^{\sqrt 2^{\sqrt 2^{\sqrt 2} }} } < 2\end{align}\]

Is this true?


Bonus: Generalize the following. \[{\LARGE \underbrace{\sqrt{x^{\sqrt{x^{\sqrt{x^{\sqrt {\cdots^{\sqrt x }}}}}}}}}_{n \text{ times}}} \]

A regular decagon is obtained by joining 10 regular pentagons side by side.

Generalizing this, we claim that there is a regular \(n\)-gon obtained by joining \( n\) regular \(k\)-gons side by side.

What is the sum of all possible values of \(k \ge 5?\)

A casino owner invents a new game where a player flips a fair coin \(n\) times in a row. If the player does not flip two heads in a row at any point in the \(n\) flips, then he wins the game; otherwise the house wins.

To make the game popular, the casino owner wants to maximize the player's chance of winning. However, the casino needs to make a profit, so the house must win more than half of the games played over the long run.

What should the casino owner set \(n\) to be?

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