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 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
An infinite plane can be tiled using identical triangles, quadrilaterals, or hexagons with no overlaps and no gaps.
Is it possible to tile an infinite plane using identical pentagons?
by
Pi Han Goh
Are you sure you want to view the solution?
A solid disk, a solid sphere, and a hoop of identical mass \(m\) and radius \(R\) are mounted at rest on frictionless axles. Each object has a string wrapped around its circumference, and the strings are each pulled with an identical force \(F\) for the same time interval; the strings do not slip.
Which object will have the greatest rotational kinetic energy at the end of the time interval?
Note: The mass of the spokes on the hoop is negligible.
by
Jim Ciccarelli
Are you sure you want to view the solution?
Lines are drawn through point \(P\) parallel to the sides of triangle \(ABC.\) The resulting 3 triangles have areas 4, 9, and 49.
What is the area of triangle \(ABC?\)
The areas of these 3 smaller triangles are 4, 9, and 49. What's the area of the big, outer triangle?
by
Gokul nath
Are you sure you want to view the solution?
In this rectangular window frame, a spider begins to weave its web. The basic framework consists of three threads, which start from points \(A, B,\) and \(C\) and are glued together at point \(D.\)
What is the length \(l_A \) of the first thread in centimeters?
Assumptions: Without tension, all threads have the same length, but they are stretched many times their original length. Thus, the tension force \(F_i \approx - k l_i \) acting along a thread is approximately proportional to its total length \(l_i \). While the lengths \(l_i \) of the threads can be different, the spring constant \(k \) is the same for all threads. Apart from the tension, no further forces act on the spider threads.
by
Markus Michelmann
Are you sure you want to view the solution?
\(a,b,\) and \(x\) are real numbers satisfying the following system of equations: \[ \begin{eqnarray} a+b &=& 3x-2 \\ ab &=& 4x^2-3x-4. \end{eqnarray} \] What is the minimum value of \( a^2+b^2\) (to three decimal places)?
by
X X
Are you sure you want to view the solution?