skip to content

STEP Support Programme

STEP Support - Assignment 8


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.

Assignment PDF: 
Hints and Partial Solutions: 

Useful Links

Underground Mathematics: Selected worked STEP questions

STEP Question database

University of Cambridge Mathematics Faculty: What do we look for?

University of Cambridge Mathematics Faculty: Information about STEP

University of Cambridge Admissions Office: Undergraduate course information for Mathematics

Stephen Siklos' "Advanced Problems in Mathematics" book (external link)

MEI: Worked solutions to STEP questions (external link)

OCR: Exam board information about STEP (external link)

AMSP (Advanced Maths Support programme): Support for University Admission Tests (external link)