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# STEP Support - Assignment 9

### Introduction

This STEP support module includes some proofs involving triangles and sketching cubics. 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 shows you how to prove some results about triangles.

The main STEP question (1993 STEP 1 Question 7) is about the graphs of cubics and how many roots the cubics have.

The final question is Euclid's proof that the base angles in an isosceles triangle are the same, known as "the pons asinorum" or "bridge of donkeys". If differs to the proof in question 1(i) in that the condition $SSS$ for congruent triangles cannot be used.

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)