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# A Summer of Maths Tweet

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1. A Summer of Maths

Hey!

1) Are you going to study Maths at Uni next year?

2) Do you want to do some extra Maths this summer?

If your answer to both questions is Yes!'', then I have the following suggestion.

We can make this a thread a place where interesting Maths problems of pre-Undergraduate/Undergraduate level are discussed.
The questions' difficulty can vary considerably, and they can be suggested by current undergraduate students, but the main focus must be on educational problems.

** I don't want people showing off with extremely difficult, not interesting and practically unsolvable with pre-Undergraduate/Undergraduate knowledge, questions. **

If you want to play this game, I have a few rules for a start.

[1] Post a question only if you believe it is of the right caliber and you have the complete solution.

[2] Indicate any required Undergraduate knowledge that people need to know in order to produce a solution to the given problem.

[3] Hints and solutions go in spoilers. In case they are required, the person who posted the question must provide them; with a reference to their origin.

List of problems offered by universities for practice during the summer.

List of suggested books that people have decided to spend time on over the summer:

Spoiler:
Show

Introductory/First-year books:

A Concise Introduction to Pure Mathematics by Martin Liebeck

Introductory Mathematics: Algebra and Analysis by Geoff Smith

First-year books:

Algebra and Geometry by Alan Beardon

Naive Set Theory by Paul Halmos

***

Mathematical Analysis book I and book II by Vladimir Zorich

Recommendations:

(Original post by TheMagicMan)
Highly... I think they're significantly better than the burkhill books which are the most common analysis books
(Original post by shamika)
Yes - they are probably the best analysis books I've come across. Annoyed I didn't know about them when I was an undergrad
(Original post by boromir9111)

***

Vector Calculus and Tensor Analysis by P. Kendall and D Bourne (my efforts to find these authors on-line were hopeless)

Schaum's Outline of Vector Analysis by Murray Spiegel.

Linear Algebra by Serge Lang

Linear Algebra by Ichirô Satake

Calculus by Michael Spivak

Calculus by James Stewart

Third-year books:

A Primer on Riemann Surfaces by Alan Beardon

A Comprehensive Introduction to Differential Geometry by Michael Spivak

List of additional resources that people suggested:

Spoiler:
Show

Linear Algebra:

Various courses:

(Original post by djpailo)
Just thought I'd add this resource:

http://tutorial.math.lamar.edu/

You can also click on any formulae to make them easier to see.

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

Cambridge Maths Tripos Part IA notes

(Original post by TwoTwoOne)
Slight plug here, I've recently uploaded my notes to (almost) all courses in Part IA, hope they are useful

http://lishen.wordpress.com/2012/07/...-maths-tripos/

The IA Groups is lectured by Saxl again this coming year I think, so these notes should be relevant enough.
Last edited by jack.hadamard; 24-07-2012 at 14:01.
2. Re: A Summer of Maths
(Original post by jj193)
...

(Original post by Lord of the Flies)
...

(Original post by TheMagicMan)
...

All BMO/IMO or AEA/STEP questions are of the right caliber.

The following is an example of a question to post.

{*} Question
Does there exist an injective function that satisfies for all ?

{**} Required knowledge

Spoiler:
Show

The definition of an injective function.

{***} Hint

Spoiler:
Show

for all .
3. Re: A Summer of Maths
(Original post by jj193)
Q1
Can you spot the mistake?

We use induction to prove that n horses are the same colour:
Base case, n=1, true.

assume true for n=k \geq 1
If we have k+1 horses. Consider the first k horses. By assumption, they are all the same colour. Take one of these k horses, and the (k+1)th horse - again, by assumption, they are the same colour.
Hence all k+1 are the same colour (here we use 'transitivity of colour' - to be formalised below)
Spoiler:
Show
The k and k+1 horse are only the same coulor from the assumption if k>2, k is not always >2 so the proof is flawed
4. Re: A Summer of Maths
A-level exams are over and you're all already effectively preparing for your uni exams, relax!
5. Re: A Summer of Maths
(Original post by jj193)
Q4 i) how can she keep her record?

Later that evening, after an unsatisfactory meal produced by Mrs Hilbert, Mr Hilbert is hungry again. Mr Hilbert eats another key from the typewriter, leaving just one key!

ii)It is possible for the receptionist to keep track of the room numbers, but how?

Spoiler:
Show

hint:
ii) there are infinitely many primes.
any natural number has a unique prime factorisation. (Fundamental Theorem of Arithmetic)
Spoiler:
Show

4 i) She could use one of the keys as a seperating character (say the keys are a and b). If room number n wants to make a call, she can put a string of n a's. the b's can be used to seperate the different numbers e.g. if 3 then 6 made a call it woud look like aaabaaaaaa. This keeps track of what room numbres have made calls.

4 ii) Using the hint, let p(k) be the kth prime. Again say there are n rooms. Say the character availible is a c. When the first room (number i) makes a call write a string of p(i) c's. When the next room (number j) makes a call multiply the number of c's written down by p(j) and make the number of c's equal to this. At any time, the number of factors of p(m) of the number of c's is the number of calls made by room m.
6. Re: A Summer of Maths
What's the difference between this thread and the TSR maths society thread?
7. Re: A Summer of Maths
All BMO/IMO or AEA/STEP questions are of the right caliber.

The following is an example of a question to post.

{*} Question
Does there exist an injective function that satisfies for all ?

{**} Required knowledge

Spoiler:
Show

The definition of an injective function.

{***} Hint

Spoiler:
Show

for all .
Spoiler:
Show
Set x=0 and x=1 gives 0>=(f(0)-0.5)^2 and 0>=(f(1)-0.5)^2 so that both f(0) and f(1) = 0.5 so f(x) is not an injection.
8. Re: A Summer of Maths
Subscribing
9. Re: A Summer of Maths
*subscribes*
10. Re: A Summer of Maths
At the Nottingham University Open Day there was a "Maths Trail" with several interesting questions on it. The questions ranged from requiring a working knowledge of arithmetic progressions and combinations to lowest common multiples and counting squares; and more emphasis was put on thinking about them than slogging through endless manipulation. I thought I'd share one with you. The question is of the kind that could be set in C1, but is an interesting one:

A rectangle is inscribed inside a circle of radius 6 units such that each of the vertices of the rectangle lie on the circumference of the circle. Given that the perimeter of the rectangle is 28 units, what is the area?

Required Knowledge:
Spoiler:
Show

- Pythagoras' Theorem

Hints:
Spoiler:
Show

What information are you given? You have both the perimeter of the rectangle, and the radius of the circle it is inscribed inside. How can you relate these in terms of the length and width of the rectangle?
Spoiler:
Show

Consider the diagonal of the rectangle

Full Solution:
Spoiler:
Show

Let X and Y represent the length and width of the rectangle. Using this and the perimeter we can automatically deduce an equation:

2X + 2Y = 28

Which simplifies to X + Y = 14

As we are told that the radius of the circle is 6, we also know that its diameter must be 12. This diameter forms the diagonal of the rectangle. From this, another equation can be deduced using Pythagoras' Theorem:

X^2 + Y^2 = 144

We now have a set of simultaneous equations to solve. Re-arranging the first to leave Y in terms of X and substituting in the second equation leaves you with this:

X^2 + (14 - X)^2 = 144

This expands to:

X^2 + X^2 - 28X + 196 = 144

Which then simplifies to:

2X^2 - 28X + 52 = 0

or:

X^2 - 14X + 26 = 0

You can then complete the square or use the quadratic formula to get the result:

X = 7 + rt(23) or X = 7 - rt(23)

Using this you can then confirm that Y has the same values (i.e. if X = 7 + rt(23) then Y = 7 + rt(23) and vice-versa)

Now you know both side lengths, you simply multiply them together to get the area. The final answer is 26.
11. Re: A Summer of Maths
(Original post by jj193)
Show that is an equivalence relation.
It's not.
12. Re: A Summer of Maths
(Original post by dbmag9)
It's not.
I agree with this; the following is a part of question on equivalence relations.

{*} Question:

Which of the following is an equivalence relation on the given set ?

a) Let with the relation .

b) Take with .

{**} Required:

Spoiler:
Show

Definition of equivalence relations and simple interpretation of the Cartesian product in the case of .
13. Re: A Summer of Maths
(Original post by Farhan.Hanif93)
What's the difference between this thread and the TSR maths society thread?
TSR Maths Society seems to be a bit more general thread than what we need.
However, I am here for the Maths, so it makes no difference to me where I will post it.

Do people prefer to post in the TSR Maths Society thread?
14. Re: A Summer of Maths
{*} Question:

Find such that for all and the inequality

is satisfied.

{**} Required:

Spoiler:
Show

The notation explicitly specifies the set of all natural numbers together with zero; i.e. the union of the natural numbers with the singleton that contains zero.
15. Re: A Summer of Maths
{*} Question:

Find such that for all and the inequality

is satisfied.

{**} Required:

Spoiler:
Show

The notation explicitly specifies the set of all natural numbers together with zero; i.e. the union of the natural numbers with the singleton that contains zero.
Spoiler:
Show

setting m=n=k=0 and re-arranging gives 0>=(f(0)-1)^2 so f(0)=1 therefore f(k)<=1 for all k. Similarly f(1)=1 so f(k)>=1 for all k. Combining these gives f(k)=k or f(x)=x as the only solution.

I like these function problems.
Last edited by james22; 01-07-2012 at 22:48.
16. Re: A Summer of Maths
(Original post by jj193)
Question:
Show that is an equivalence relation.
I struggled with this part.
17. Re: A Summer of Maths
(Original post by james22)
Spoiler:
Show

Combining these gives f(k)=k or f(x)=x as the only solution.
Spoiler:
Show

If , then what value should take?
18. Re: A Summer of Maths
(Original post by Bobifier)
I struggled with this part.
19. Re: A Summer of Maths
(Original post by james22)
Spoiler:
Show

setting m=n=k=0 and re-arranging gives 0>=(f(0)-1)^2 so f(0)=1 therefore f(k)<=1 for all k. Similarly f(1)=1 so f(k)>=k for all k. Combining these gives f(k)=k or f(x)=x as the only solution.

I like these function problems.
Having , does not yield

Also, you stated which negates your conclusion.
20. Re: A Summer of Maths
{*} Question:

Find such that for all and the inequality

is satisfied.

{**} Required:

Spoiler:
Show

The notation explicitly specifies the set of all natural numbers together with zero; i.e. the union of the natural numbers with the singleton that contains zero.
f(x)=1 for all the x we're allowed. done.

Spoiler:
Show
you only said find an f