Topic 0: IntroductionDr J Frost (jfrost@tiffin.kingston.sch.uk)

Last modified: 12th September 2014

? Any box with a ? can be clicked to reveal the answer (this works particularly well with interactive whiteboards!).Make sure you’re viewing the slides in slideshow mode.

A: London B: Paris C: Madrid

For multiple choice questions (e.g. SMC), click your choice to reveal the answer (try below!)

Question: The capital of Spain is:

Slide Guidance

What and who this club is for

• Senior Maths Challenge• British Maths Olympiad• MAT• A little bit of STEP• Giving a general broad grounding

of maths to help with university interviews, etc.

What we’ll concentrate on

• People who enjoy maths who want to explore the more interesting aspects outside of the A Level curriculum.

• People who want to do well in the SMC/BMO.

• People who are considering entry for a maths-related degree at a competitive university (particularly Oxbridge).

• I don’t presume people are doing Further Maths.

Who this is for

This is not intended to help you get an A* at A Level or supplement the content you do in your usual classes. Much of what we’ll cover (elementary number theory, combinatorics, functional equations) will likely be completely new topics to you.

What this is not…

Format/Admin• Please ensure you put your name, email address and

school on the sign up sheet before you go.• Please turn up on time!• Most sessions you will be given a problem sheet to take

home with you. You’ll be emailed solutions the day before the next session. It’s your responsibility to look through these solutions; we won’t be going over them in the sessions (although you’re welcome to ask me questions on them)

• While these won’t be marked, please put as much effort into them as possible – it doesn’t matter if you can’t answer all the questions. ‘Doing problems’ is the most effective way of appreciating the content we’ve covered.

To keep organised, you should put together a folder with dividers. Appropriate headings could be:1. Geometry 2. Number Theory 3. Algebra 4. Combinatorics5. Graphs 6. Prob/Stats 7. Functional Equations8. Logic 9. Pigeonhole Principle 10. MAT 11. Other

Website

www.drfrostmaths.com/rzc

All slides, problem sheets, solutions and other supplementary materials will be put on this website. Some topic slides will have an ‘Epilogue’ at the end: slides we won’t have time to go through in these sessions, but are available for you to view online.

MAT• This is the Oxford ‘Maths Admissions Test’.• Up until 2013, was used exclusively by Oxford for those applying for Maths (and

joint courses) or Computer Science (and joint courses).• Imperial College now use it for maths applicants.• 2.5 hour (non-calculator) test. 40% of marks from multiple choice questions, 60%

from longer questions.• Based on the C1 and C2 A Level Maths modules. But these aren’t just ‘harder A

Level questions’: they require a degree more ingenuity and dexterity.• But it is the kind of test you can prepare for to some extent, and we’ll explore

some tips and theory useful for these papers later in the year.• At Oxford, a mark of roughly 50% required to be invited to interview, and the

average mark of a successful applicant is about 75%.

British Maths Olympiad• A 3 ½ hour paper consisting of just 6 questions.• You’re invited to take it if you’re in roughly the top 1300 scorers in the Senior

Maths Challenge (in 2012, you needed a score of 93 out of 125 – this corresponded to getting 17 out of 25 questions right)

• These questions are very difficult and the challenge is often knowing where to start. Many of the questions however become more approachable if you’ve had prior exposure to questions on a similar topic.

• Some questions are more ‘routine’ if you know the typical strategies. Others require a novel idea that might not be obvious to think up.

• There’s a ‘Distinction’ award if you’re within the top 25% of people taking the test.• The top 100 are awarded prizes, and most of these people are invited to ‘BMO

Round 2’. A small handful of these are subsequently selected for the national team to compete in the International Maths Olympiad.

Senior Maths Challenge/BMO• The Senior Maths Challenge is on November 7th 2013.• Hence we’ll start with the topics that will be most useful for this test: Geometry,

Number Theory and Algebra.• The BMO is shortly after.

Geometry50%

Algebra20%

Number Theory

19%

Other9%

Probability3%

SMC Topics

This was the makeup of ‘harder’ questions (Q16-25) on the SMC papers from 2004-2012.

It goes without saying that you should concentrate your efforts on Geometry!

STEP• This exam paper is used as an offer requirement for Cambridge and Warwick mathematics

courses. In contrast to MAT, you would take this in the summer of Year 13, not prior to your university interview.

• The paper is 3 hours. You have to choose 6 of the 13 questions (8 pure, 3 mechanics, 2 probability and statistics).

• STEP paper questions are in some ways just ‘hard A Level questions’. But they differ in a few ways:• They are less guided than A Level questions. e.g. You might have to integrate by an

appropriate substitution: in an A Level exam they would give you the substitution, in a STEP paper they wouldn’t.

• They require a good deal more technical dexterity – you can’t afford to be sloppy in your maths.

• Contrary to some people’s perceptions, STEP I, STEP II and STEP III are of comparable difficulty (and indeed a similar distribution of people get the S, 1, 2 grades in each). It’s just that later STEP papers are based on material later into the Maths/Further Maths A Level syllabus.

• In contrast, MAT and BMO doesn’t really require any knowledge beyond C1 and C2.

Different exams

The extent to which the style of question deviate from A Level ones.

Diffi

culty

A Level

MAT

STEP

BMO

(This is a very rough judgement!)

Topics we’ll cover across the year

Geometry

Number Theory

e.g. “How many multiples of 2013 have 2013 factors?”

Algebra

e.g. Prove that for all real and .

Combinatorics

e.g. How many ways of pairing people at a party?

Graph Sketching/Reasoning

e.g. Sketch

Probability/Stats

e.g. If a throw a needle of length onto a board with horizontal lines a distance of apart, what’s the probability the needle crosses a line?

Functional Equations

e.g. Find all functions which satisfy .

Pigeonhole Principle

e.g. A cinema sells 31 tickets in total one evening, and each movie can be watched by up to 3 people. Show that the minimum number of movies seen by the same number of people is 6.

Logic

e.g. What is the maximum number of the following statements that can be true? …

MAT Prep

ζSet NotationIntroduction

Here, we cover some basic set notation and common sets used to represent collections of objects (usually numbers).

What is a set?

A set is a collection of items (often numbers) with two properties:

1. It can’t contain duplicates.2. The order of the items doesn’t matter.

We can use curly braces to define a set, e.g. By (2) above, .If a set has no items, it is known as the empty set, denoted by the symbol: .

We say an item is a ‘member’ of a set if it is contained within the set.If is a set, then:

means that is a member of .

2∈ {2,4 }{1 }∈ {1,2,3 }

False True

False True

(Click to vote!)

This is like asking “in a bag containing balls labelled 1 and 2, do we have within this bag another bag containing a 1?” No we don’t!

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Infinite SetsSome sets are infinite in size. Some are so common we use special symbols to represent them:

Symbol Name

Natural numbers It’s not generally agreed whether 0 is included. We could use if we wanted to make explicit the 0 was included, and to mean just ‘positive numbers’ (which excludes 0).

Integers

Rational numbersAll numbers which can be expressed as fractions , where and are integers.

Real numbersIncludes all numbers which can’t be expressed as fractions.

Complex numbers

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Infinite Sets

ℝℚ

ℤℕ

4-3

3.4 𝟔𝟕

𝟒 . �̇�𝝅 √𝟓 √−𝟏

ℂClick to boxes to move them to the correct set in this Venn Diagram.

Infinite Sets

Set notation is particularly useful when talking about functions:

for all real .

for all .

, . means “for all”

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ζProofIntroduction

We look at what constitutes a ‘proof’, when a proof might be incomplete, and the different types of proof often available.

When a proof is not a proof

Egbert claims that is prime for all integers .He tries it for from 1 to 39 and finds that the result is prime every time.

How many values of would we need to try to show Egbert is wrong?Just the one. This is known as a counterexample.

How many values of would we need to try to show Egbert is right?Infinitely many, and clearly it wouldn’t be theoretically possible to show the statement is true by exhaustively trying every . We’d need some alternative method.Just trying is not sufficient to prove the statement, and indeed, the statement is false.

For example, mathematicians have found millions of pairs of primes which differ by 2, which would at most ‘hint’ there’s infinitely many, but no one has yet been able to formally ‘prove’ that there are infinitely many twin primes.

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Counterexamples

A B C D

Source: MAT

As we’ll see in the Logic topic later this year, the statement “If , then ” is only false when is true but is false.

What’s wrong with these proofs?

She’s given an upper bound for the maximum (i.e. that the maximum can’t be more than 32), but hasn’t actually shown we give an arrangement of 32 counters which satisfies the conditions. For all we know, it’s not possible to do it with 32 counters.

Edwina: “Each row can have up to 4 counters in it. There’s 8 rows, so clearly we can’t have more than 32 counters.”

He hasn’t justified that we can’t have more than 32 counters.This is easy to show, but needs to be explicitly stated.

Bob: Bob simply draws a valid arrangement using 32 counters(and let’s say he’s right: the maximum is indeed 32 counters)

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BMO1

What’s wrong with these proofs?

He hasn’t shown than 10 doesn’t work, i.e. that 11 is the smallest .Dave: Dave manages to (correctly) identify that that works.

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BMO1

Proof by contradiction

Proofs by contradiction are quite common in the BMO, so it’s always worth having at the back of your mind to look out for sensible places to use it.

So for this problem, to show satisfies the condition given:Assume it doesn’t satisfy the condition, i.e. that there exists a such that . Show that this leads to a contradiction.You could break the problem down by considering whether the first few numbers are the same or different colours, and then thinking what colouring the later numbers must have to avoid satisfying the condition.

In general, to prove a statement by contradiction, we… assume the statement is false, and show this leads to a contradiction.?

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Proof by contradiction

How would we use a proof by contradiction here?

Assume there was two distinct places to dig the treasure, and . Then we’d need to show that in fact , i.e. , i.e. a contradiction, and thus there is only one place to dig.

Doing so would involve forming 4 ‘Diophantine Equations’ (equations for which we’re looking for integer solutions) which we’ll cover in the Number Theory topic.

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BMO1

Two-way proofs

BMO1

Sheila: Sheila successfully manages to show that if , then .What’s wrong with her proof?

She only proved the statement in ‘one direction’. She didn’t prove that “if then . If we have the condition, “If is true then true”, then it’s quite possible that can be true when is false, because the condition only cares about the case when is true, not when is false.

But “If, and only if is true, is true” says something more. Suppose was true. Then unlike above, can’t be false, because we said was true only if was true. Thus if is true, is true. So:

X is true if, and only if, Y is true.

Proofs needed:

1. If X is true, then is true.2. If is true, then is true.

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ζEpilogue: Proof by InductionIntroduction

(‘Epilogue’ slides are not covered in class)

Proof by InductionFurther Mathematicians will do this soon enough in FP1, but for the benefit of those not doing this A Level course, here’s a quick intro.

Suppose I wanted to show that some expression involving had some property for all possible natural number values of , e.g.

Prove that is divisible by 5 for all

Clearly we can’t exhaustively check every value of up to infinity! But suppose we showed that:

a. It’s true when b. If it’s true for , then it’s true for

If we’ve showed it’s true when , then by (b), it’s also true for . And if it’s true for , again by (b) it’s true for . And hence we would have shown it’s true for all natural numbers.

This is known as the base case.

This is known as the inductive step.

Proof by Induction

Prove that is divisible by 5 for all

Base case:When , then is divisible by 5.

Inductive step:Assume the it’s true for . Then show it’s true for :

Clearly is divisible by 5. And because we assumed the statement is true for , is divisible by 5 as well. So we’ve shown that if it’s true for , it’s true for .

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Proof by InductionThere’s a slight variant of proof by induction which you WON’T see in FP1:(there’s another good example in the Functional Equations slides)

a. It’s true when b. If it’s true for , then it’s true for

If it’s true for then it’s true for

Example:

Prove that every positive integer greater than one has at least 1 prime divisor.

Base case:When , then 2 itself is prime, and has itself as a prime divisor.

Inductive step:Assume the it’s true for . If is prime, then it has itself as a prime divisor.If it’s not prime, i.e. composite, then for some integers and .But and . By induction, both are divisible by some prime divisor, say and . Then will be divisible by and .

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