### Shades of Fermat's Last Theorem

The familiar Pythagorean 3-4-5 triple gives one solution to (x-1)^n + x^n = (x+1)^n so what about other solutions for x an integer and n= 2, 3, 4 or 5?

### Exhaustion

Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2

### Code to Zero

Find all 3 digit numbers such that by adding the first digit, the square of the second and the cube of the third you get the original number, for example 1 + 3^2 + 5^3 = 135.

# Inner Equality

##### Age 16 to 18 ShortChallenge Level

Inequalities are an important extension of algebra which are needed more formally in C1 and beyond.

Note that some, but not all, algebraic manipulations still work with equals signs 'replaced' by inequality signs -- you need to take extra care when algebraically manipulating inequalities.

Addition or subtraction of a quantity is straightforward with inequalities. For example, if we know that $5< a+b< 10$ then we know that $5-b< a< 10-b$. However, we need to take more care with division and multiplication as minus signs cause inequalities to reverse under these operations.

Direct algebra will not help you much in this problem. You will have to make deductions such as 'if a is a very small positive number than b must be very close to 5'.

Writing down such statements is difficult to do clearly, so focus on the inequalities intuitively if need be.