### Introduction

This STEP support module introduces the AM-GM inequality and the intersections of circles and ellipses. Previous assignments can be found here, but you can do this one without having done the others first.

STEP questions are difficult, they are supposed to be and you should expect to get stuck. However, as you tackle more and more STEP questions you will develop a range of problem solving skills (and spend less time "being stuck").

### About this assignment

The assignment is published as a pdf file below. Each STEP Support assignment module starts with a warm-up exercise, followed by preparatory work leading to a STEP question. Finally, there is a warm-down exercise.

The **warm up** for this assignment proves the AM-GM inequality for 2, 3 and 4 values. The AM-GM inequality states that the arithmetic mean of a set of values is greater than or equal to the geometric mean of those values and is a very useful and powerful result.

The **main STEP question (2002 STEP 1 Question 1)** is about the intersections of a circle and an ellipse; you don't need to know anything about ellipses to tackle the question but once you have done it you might like to visit Desmos to draw the graphs and see what is happening.

The **final question** revisits the "Socks" problem first seen in assignment 5, but this time there are three colours to consider.

A nice result connecting the arithmetic mean, the geometric mean and the harmonic mean is discussed here .

There is a discussion of some of the different ways of proving the $n$ value case here .

Ellipses and other "conic sections" are discussed in this Plus maths article.

Some more hints and tips for solving Geometry STEP questions can be found in this Advanced Problem Solving module.

### Hints, support and self evaluation

The **“Hints and partial solutions for Assignment 8”** file gives suggestions on how you can tackle the questions, and some common pitfalls to avoid, as well as some partial solutions and answers.