# Problems of the Week

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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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