The Student Room Group

A Summer of Maths

Poll

Have you studied any Group Theory already?

Please, do not ask A-level questions. :smile:


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.

Spoiler



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

Spoiler




List of additional resources that people suggested:

Spoiler



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

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 :smile:

http://lishen.wordpress.com/2012/07/16/some-of-my-notes-for-the-part-ia-cambridge-maths-tripos/

The IA Groups is lectured by Saxl again this coming year I think, so these notes should be relevant enough.
(edited 11 years ago)

Scroll to see replies

Reply 1
Original post by jj193
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Original post by Lord of the Flies
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Original post by TheMagicMan
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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 f:RRf:\mathbb{R}\to\mathbb{R} that satisfies f(x2)[f(x)]214\displaystyle f(x^2) - [f(x)]^2 \geq \frac{1}{4} for all xx?

{**} Required knowledge

Spoiler



{***} Hint

Spoiler

Reply 2
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

Reply 3
A-level exams are over and you're all already effectively preparing for your uni exams, relax!
Reply 4
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



Spoiler

What's the difference between this thread and the TSR maths society thread? :confused:
Reply 6
Original post by jack.hadamard
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 f:RRf:\mathbb{R}\to\mathbb{R} that satisfies f(x2)[f(x)]214\displaystyle f(x^2) - [f(x)]^2 \geq \frac{1}{4} for all xx?

{**} Required knowledge

Spoiler



{***} Hint

Spoiler



Spoiler

Reply 7
Subscribing :ninja:
*subscribes*
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



Hints:

Spoiler



Full Solution:

Spoiler

Reply 10
Original post by jj193
Show that \geq is an equivalence relation.

It's not.
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 SS?

a) Let S=NS = \mathbb{N} with the relation ab    ab is a squarea \sim b \iff ab\ \text{is a square}.

b) Take S=R×RS = \mathbb{R} \times \mathbb{R} with (x,y)(a,b)    x2+y2=a2+b2(x,y) \sim (a, b) \iff x^2 + y^2 = a^2 + b^2.


{**} Required:

Spoiler

Original post by Farhan.Hanif93
What's the difference between this thread and the TSR maths society thread? :confused:


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?
{*} Question:

Find f:N{0}Rf : \mathbb{N} \cup \{0\} \to \mathbb{R} such that for all k,mk,m and nn the inequality

f(km)+f(kn)f(k)f(mn)  1f(km) + f(kn) - f(k)f(mn)\ \geq\ 1

is satisfied.


{**} Required:

Spoiler

Reply 14
Original post by jack.hadamard
{*} Question:

Find f:N{0}Rf : \mathbb{N} \cup \{0\} \to \mathbb{R} such that for all k,mk,m and nn the inequality

f(km)+f(kn)f(k)f(mn)  1f(km) + f(kn) - f(k)f(mn)\ \geq\ 1

is satisfied.


{**} Required:

Spoiler



Spoiler



I like these function problems.
(edited 11 years ago)
Reply 15
Original post by jj193

Question:
Show that \geq is an equivalence relation.


I struggled with this part.
Original post by james22

Spoiler




Spoiler

Reply 17
Original post by Bobifier
I struggled with this part.


:colondollar:
Original post by james22

Spoiler



I like these function problems.


Having f(k)1f(k)\leq1, f(k)kf(k)\geq k does not yield f(x)=xf(x)=x

Also, you stated f(0)=1f(0)=1 which negates your conclusion.
Reply 19
Original post by jack.hadamard
{*} Question:

Find f:N{0}Rf : \mathbb{N} \cup \{0\} \to \mathbb{R} such that for all k,mk,m and nn the inequality

f(km)+f(kn)f(k)f(mn)  1f(km) + f(kn) - f(k)f(mn)\ \geq\ 1

is satisfied.


{**} Required:

Spoiler



f(x)=1 for all the x we're allowed. done.

Spoiler

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