### Surds

Find the exact values of x, y and a satisfying the following system of equations: 1/(a+1) = a - 1 x + y = 2a x = ay

### The Root of the Problem

Find the sum of the series.

### Fit for Photocopying

Photocopiers can reduce from A3 to A4 without distorting the image. Explore the relationships between different paper sizes that make this possible.

# Origami Shapes

##### Stage: 4 Challenge Level:

The first thing to notice when solving this problem is that the sides of a sheet of A4 paper are in the ratio $1$ to $\sqrt{2}$ . (Check this by taking a sheet and measuring it.)

It simplifies the working a great deal if we treat the sheet of A4 paper as a rectangle with sides $1$ and $\sqrt{2}$, and keep the working exact by leaving the square roots in our expressions rather than typing them into a calculator.