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 6 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
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 6 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Problems of the Week
Contribute a problem- Week of December 10
- Week of December 3
- Week of November 26
- Week of November 19
- Week of November 12
- Week of November 5
- Week of October 29
- Week of October 22
- Week of October 15
- Week of October 8
- Week of October 1
- Week of September 24
- Week of September 17
- Week of September 10
- Week of September 3
- Week of August 27
- Week of August 20
- Week of August 13
- Week of August 6
- Week of July 30
- Week of July 23
- Week of July 16
- Week of July 9
- Week of July 2
- Week of June 25
- Week of June 18
- Week of June 11
- Week of June 4
- Week of May 28
- Week of May 21
- Week of May 14
- Week of May 7
- Week of April 30
- Week of April 23
- Week of April 16
- Week of April 9
- Week of April 2
- Week of March 26
- Week of March 19
- Week of March 12
- Week of March 5
- Week of February 26
- Week of February 19
- Week of February 12
- Week of February 5
- Week of January 29
- Week of January 22
- Week of January 15
- Week of January 8
- Week of January 1
- Week of December 25
- Week of December 18
- Week of December 11
- Week of December 4
- Week of November 27
- Week of November 20
- Week of November 13
- Week of November 6
- Week of October 30
- Week of October 23
- Week of October 16
- Week of October 9
- Week of October 2
- Week of September 25
- Week of September 18
- Week of September 11
- Week of September 4
- Week of August 28
- Week of August 21
- Week of August 14
- Week of August 7
- Week of July 31
- Week of July 24
- Week of July 17
- Week of July 10
- Week of July 3
- Week of June 26
- Week of June 19
- Week of June 12
- Week of June 5
- Week of May 29
- Week of May 22
- Week of May 15
- Week of May 8
- Week of May 1
- Week of April 24
- Week of April 17
- Week of April 10
- Week of April 3
- Week of March 27
- Week of March 20
- Week of March 13
- Week of March 6
- Week of February 27
- Week of February 20
- Week of February 13
- Week of February 6
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
Are you sure you want to view the solution?
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
Are you sure you want to view the solution?
\[\LARGE \begin{align}\large \sqrt{2^{\sqrt{2^{\sqrt{2^{\sqrt {2^{\sqrt 2}}}}}}}} = \sqrt 2^{\sqrt 2^{\sqrt 2^{\sqrt 2^{\sqrt 2} }} } < 2\end{align}\]
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
Are you sure you want to view the solution?
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
Are you sure you want to view the solution?
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
Are you sure you want to view the solution?