This is a wonderful website offering a long list of interesting and challenging math and computer science problems.
There are many different kinds of problems to solve: some are purely mathematical in nature, and some would be considered more like pure computer science.
Some are easy (“What is the 1001st prime number?“), some seem moderately challenging (“What is the sum of the digits of 
The problems are freely available, but it looks like you have to register to submit answers.
Here’s a nice little quiz on plotting points in the Cartesian plane:
http://www.thatquiz.org/tq/practice.html?idpoints-j8-l5-m2kc0-na-p0
After you’ve warmed up, be sure to try plotting complex numbers, or test your knowledge of the Polar plane (in both degrees and radians).
There are a number of simple little quizzes to enjoy at thatquiz.org; just be careful not to let the day slip away!
There is a great set of videos from MathsCareers.org.uk that focus on mathematics in industry. The videos speak to strong math students who have questions about finding careers that utilize their mathematical abilities.
The speakers are employers and recent graduates working in technical fields, and they offer a lot of clear, thoughtful advice.
The “Why are ‘Soft Skills’ Important?” segment is particularly great to see: the speakers all stress how communcation, writing, presenting, and teamwork skills are crucial to success in technical fields. In addition, the “Why are Maths Graduates Sought After?” segment really highlights what is fundamentally important in mathematics: problem solving skills, thinking creatively, and the ability to break complex ideas down into simpler ones. These skills are prized in every field and industry.
The best advice I saw throughout the videos is to start thinking early about what really interests you and explore all the options you can. There’s more to math than just teaching and banking!
This is an engaging, accessible, and surprisingly moving documentary about Andrew Wiles and his lifelong pursuit of Fermat’s Last Theorem:
http://video.google.com/videoplay?docid=8269328330690408516
Although the mathematics of the proof could not possibly be explained to the layperson (there aren’t many people in the world who could really understand it in its entirety), this BBC documentary does a great job of narrating the struggles, setbacks, and triumphs of Wiles’ pursuit.
The story of the hero and the many peripheral characters (including John Conway) opens a wonderful window into the world of advanced mathematics.
This is an amazing resource from the British Columbia Institute of Technology: a cross-referenced list of how various mathematical ideas are used in various technology fields.
http://commons.bcit.ca/math/examples/
Interested in how Linear Algebra can be applied to Nuclear Medicine? Or how Logs and Exponentials can be applied to Forestry and Wildlife? Or how Differential Equations can be applied to Mechanical Engineering?
Well, look no further!