1. If a, b, c are positive real numbers and log(a, b) + log(b, c) + log(c, a) = 0, find the value of (log(a, b))^3 + (log(b, c))^3 + (log(c, a))^3

2. Find log(n, 1/2) * log(n-1, 1/3) * … * log(2, 1/n) in terms of n.

NB log(a, x) means log base a of [...]

Define the operator * such that, a*b = ab + a + b for all integers a and b.

Evaluate 1*(2*(3*(4*…(99*100)…)))

NB ab means a times b in the usual sense. The title is a clue.

Here’s another: you need to “invert” the order of the expression.

What is the remainder when (x + 1)^2010 is divided by x^2 + x + 1 ?

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

A mother is 21 years older than her child. In exactly 6 years from now, the mother will be exactly 5 times as old as the child. Where’s the father?

There are two red, two black, two white, and some blue socks in a drawer. If two socks are taken from the drawer (without replacement), the probability they are of the same color is 1/5. How many blue socks are there in the drawer?

A cat is going up a stairway with thirteen steps. However, instead of walking up the stairs one step at a time, the cat jumps two, three or four steps at a time (though if necessary, it will just walk the last step). How many different ways can the cat go from the bottom to [...]

Matt has somewhere between 1000 and 2000 pieces of paper he’s trying to divide into piles of the same size (but not all in one pile or piles of one sheet each). He tries 2, 3, 4, 5, 6, 7, and 8 piles but ends up with one sheet left over each time. How many [...]

An unfair coin has the property that when flipped four times, it has the same probability of turning up 2 heads and 2 tails (in any order) as 3 heads and 1 tail (in any order). What is the probability of getting a head in any one flip?

How many positive integers n < 1000 makes the following be prime?:

3^n + 5^n + 7^n + 11^n + 13^n + 17^n + 19^n