Problems of the Week

Contribute a problem

2018-09-17 Intermediate

         

A number of the form 10n1=9999...9n times,10^n-1=\underbrace{9999...9}_{n\text{ times}}, where nn is a positive integer, will never be divisible by 22 or 5.5.

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?

22222=22222<2\LARGE \begin{aligned}\large \sqrt{2^{\sqrt{2^{\sqrt{2^{\sqrt {2^{\sqrt 2}}}}}}}} = \sqrt 2^{\sqrt 2^{\sqrt 2^{\sqrt 2^{\sqrt 2} }} } < 2\end{aligned}

Is this true?


Bonus: Generalize the following. xxxxn times{\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 nn-gon obtained by joining n n regular kk-gons side by side.

What is the sum of all possible values of k5?k \ge 5?

A casino owner invents a new game where a player flips a fair coin nn times in a row. If the player does not flip two heads in a row at any point in the nn 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 nn to be?

×

Problem Loading...

Note Loading...

Set Loading...