Excel in math and science
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Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, interactive explorations.
Used and loved by over 7 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
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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?
by
Mike Pannekoek
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A circle is inscribed in a right trapezoid with base lengths 22 and 35.
What is the area of this trapezoid?
by
Khang Nguyen Thanh
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\[\LARGE \begin{align}\large \sqrt{2^{\sqrt{2^{\sqrt{2^{\sqrt {2^{\sqrt 2}}}}}}}} = \sqrt 2^{\sqrt 2^{\sqrt 2^{\sqrt 2^{\sqrt 2} }} } < 2\end{align}\]
Is this true?
Bonus: Generalize the following. \[{\LARGE \underbrace{\sqrt{x^{\sqrt{x^{\sqrt{x^{\sqrt {\cdots^{\sqrt x }}}}}}}}}_{n \text{ times}}} \]
by
Naren Bhandari
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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 \(k \ge 5?\)
by
Chan Lye Lee
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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?
by
David Vreken
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