Given that  \[x>0\], solve the equation\[\displaystyle \frac{4^{x^2}}{8^x} = 4\]

An author has grown tired of writing books. Instead, he decides to take a large amount of paper and, on each piece of paper, write down a different string of 100 characters using a set of 25 characters (22 letters, a space, a comma and a period).
Let P be the number of pieces of paper he must use if he is to write down every possible string of 100 characters from his set of 25 characters.
How many digits are in P?      (You may use the fact that \[log_{10}(2) \approx 0.3\])

The number \[371\] has only odd digits, since \[3\],\[7\] and \[1\] are all odd.
The number \[493\] does not have only odd digits, since \[4\] is even.
The number \[339\] has only odd digits, but the number \[3\] is repeated.
How many numbers \[n\] with  \[1\leq n<10000\]  have only odd digits such that no digits are repeated?

Consider the numbers formed by all possible permutations of the numbers in the set \[\{1,2,3,4\}\]:
e.g. \[1234,1243,1324,\ldots,4321\]
What is the sum of all these numbers?

How many (real) solutions can be found to the equation
\[\displaystyle (x^2 -5x+5)^{(x^2-11x+30)} = 1\]?

Solve the equation \[x-1+x-2+x-3 + \cdots + x-99 = 0\]

The functions \[f(x)\] and \[g(x)\] satisfy \[ \displaystyle \begin{array}{lcr} f(x) + g(x) & = & 2x-5 \\ f(x) - g(x) & = & 1-3x\end{array}\]
Find the value of \[2\cdot f(1)\cdot g(0)\]

There are 44 sweets in a jar. All sweets are either blue or green and there are more green sweets than blue.
Two sweets are randomly picked and the probability that they are different colours is \[\displaystyle\frac{192}{473}\]. 
How many blue sweets are there?

How many pairs of integers \[m,n\]  with  \[1 \leq m \leq n \leq 2017 \] satisfy the equation \[m^2+n^2=m^3\]?

The roots and vertex of the parabola with equation \[ y=100x^2−10000\] form a triangle whose area may be represented in the form \[10^n\].
What is the value of \[n\]?