A group of 10 people are seated at a circular table. They hadn’t noticed that they had places designated for them, and none of them sat in the right place. Prove that simply by rotating the table, that at least 2 of them can be sat in the right place.

What is the smallest factor of 123456! + 1?

That number is approximately 2.6 * 10574964

I’ve purloined this from the old site.

A table mat is made from red and white beads. There is an inner rectangle made entirely using white beads. It has a border (on all four sides) made from a single layer of red beads. Which mats can have as many white beads as red beads?

All the beads [...]

Evaluate Sum(n = 0 to ∞, cos(n x)/2^n), where cos(x) = 1/5.

Indirect hint, really an excuse to point out the remarkable proof of a distant cousin to the above problem, see: http://en.wikipedia.org/wiki/Gaussian_integral

Give the formula for the coordinates of the intersecting rectangles formed by two separate rectangles in 2D space ?

An investor puts $10,000 into a weekly trading scheme.

In each week it is equally likely that he’ll make an 80% gain or a 60% loss (there are no other possibilities). One year later (52 weeks), roughly how much money does the investor expect to have? The answer is one of the following:

a) $1.95, b) $14,000, c) $140,000, [...]

Prove that for any natural number that ends in 1, 3, 7 or 9, that it has an infinite number of multiples which consists only of a string of 1’s.

e.g. 13*8547 = 111111

NB It is trivial to show that numbers ending in 0, 2, 4, 5, 6 or 8 cannot possses that property.

A boy, a girl and a dog go for a 10 mile walk. The boy and girl can walk at 2 mph and the dog can trot at 4 mph. They also have a bicycle which only one of them including the dog (it used to work in a circus, OK) can use at a [...]

Four bugs are at the four corners of a square of side length D. They start walking at constant speed in an anticlockwise direction at all times directly towards the bug ahead of them.

How far does each bug walk before they meet with each other?

A group of children share marbles from a bag. The first child takes one marble and a tenth of the remainder. The second child takes two marbles and a tenth of the remainder. The third child takes three marbles and a tenth of the remainder. And so on until the last child takes whatever is [...]