The Student Room Group

Difficult Maths/Physics Problems Help Thread

This is a thread for helping each other solve and understand interesting/difficult problems in maths and physics. A good source of these is from the MAT, PAT, STEP and Olympiad exams. Here are some useful sites:

PAT papers: https://www2.physics.ox.ac.uk/study-here/undergraduates/applications/physics-aptitude-test-pat/pat-past-papers
PAT solutions: http://www.physicsandmathstutor.com/pat/
MAT papers: http://www.maths.ox.ac.uk/study-here/undergraduate-study/maths-admissions-test
Physics Olympiad papers/solutions: http://www.physics.ox.ac.uk/olympiad/PastPapers.html
Maths Olympiad papers: http://www.bmoc.maths.org/home/bmo.shtml
STEP papers/solutions: http://www.mathshelper.co.uk/oxb.htm
Engineering problems: http://i-want-to-study-engineering.org/

The idea is that by assisting others we'll all become more familiar with these types of questions which will help in university admissions tests and problem solving ability. Plus, they're fun to solve. Feel free to post questions you've already done and want to share.

Please put full solutions to questions people post in 'spoiler' tags by writing [SPOILER ] text [/SPOILER ] (without spaces).
(edited 9 years ago)

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Original post by Elcor
This is a thread for helping each other solve and understand interesting/difficult problems in maths and physics. A good source of these is from the MAT, PAT, STEP and Olympiad exams. Here are some useful sites:

PAT papers: https://www2.physics.ox.ac.uk/study-here/undergraduates/applications/physics-aptitude-test-pat/pat-past-papers
PAT solutions: http://www.physicsandmathstutor.com/pat/
MAT papers: http://www.maths.ox.ac.uk/study-here/undergraduate-study/maths-admissions-test
Physics Olympiad papers/solutions: http://www.physics.ox.ac.uk/olympiad/PastPapers.html
STEP papers/solutions: http://www.mathshelper.co.uk/oxb.htm

The idea is that by assisting others we'll all become more familiar with these types of questions which will help in university admissions tests and problem solving ability. Plus, they're fun to solve. Feel free to post questions you've already done and want to share.


you can also link the british maths olympiad website aswell. don leave us hanging πŸ˜‰


Posted from TSR Mobile
Ill post the first question.

x+y=xy=3
find x^3 +y^3 with proof.
nat sci question.


Posted from TSR Mobile
Original post by physicsmaths
Ill post the first question.

x+y=xy=3
find x^3 +y^3 with proof.
nat sci question.


Posted from TSR Mobile


Spoiler



There's a slicker way isn't there?
Reply 4
Original post by physicsmaths
you can also link the british maths olympiad website aswell. don leave us hanging


Posted from TSR Mobile


I added it. :redface:

Did you sit the Senior Maths Challenge? I did and (barely) got a gold haha
Reply 5
Original post by physicsmaths
Ill post the first question.

x+y=xy=3
find x^3 +y^3 with proof.
nat sci question.


Posted from TSR Mobile

Spoiler

Reply 6
A quick starter question that my physics teacher once gave me (don't google it - ruins the fun)

Show that ln(n!) = nln(n) - n + 1

There is a much more rigorous approach to this than what I used (but who knows just how smart some of you are) but it doesn't require anything beyond core modules.
Reply 7
Original post by L'Evil Fish

Spoiler



There's a slicker way isn't there?


Spoiler


Posted from TSR Mobile
Reply 8
Original post by physicsmaths
Ill post the first question.

x+y=xy=3
find x^3 +y^3 with proof.
nat sci question.


Posted from TSR Mobile


Original post by L'Evil Fish

Spoiler



There's a slicker way isn't there?


I also got 0
Reply 9
Original post by L'Evil Fish

Spoiler



There's a slicker way isn't there?


Original post by Red Fox

Spoiler



I essentially did the same thing, can't think of a more elegant way to prove it though
Original post by Navo D.
A quick starter question that my physics teacher once gave me (don't google it - ruins the fun)

Show that ln(n!) = nln(n) - n + 1

There is a much more rigorous approach to this than what I used (but who knows just how smart some of you are) but it doesn't require anything beyond core modules.


Ignore. Lol

Which side did you start from hmm

Spoiler


Bleurgh
(edited 9 years ago)
Reply 11
Actually I like the idea of putting solutions in spoilers, so someone attempting a problem doesn't accidentally see the first step or two, which as you know are often the hardest parts of a problem.

How do you make a spoiler again?
Reply 12
Original post by Elcor
Actually I like the idea of putting solutions in spoilers, so someone attempting a problem doesn't accidentally see the first step or two, which as you know are often the hardest parts of a problem.

How do you make a spoiler again?



[ SPOILER ] text goes here [ /SPOILER ]

remove the spaces
Reply 13
Original post by L'Evil Fish
Ignore. Lol

Which side did you start from hmm


LHS; try expanding it and see if something clicks
Original post by Elcor
I added it. :redface:

Did you sit the Senior Maths Challenge? I did and (barely) got a gold haha


yes i did it for the second time. i got lower this time 😝


Posted from TSR Mobile
Thanks for making this thread. Should be fun if it stays active for a bit...


Original post by physicsmaths
Ill post the first question.

x+y=xy=3
find x^3 +y^3 with proof.
nat sci question.




Spoiler

Original post by L'Evil Fish

Spoiler



There's a slicker way isn't there?


nope its perfecto. well thas how i did it aswell.


Posted from TSR Mobile
Reply 17
Original post by Necrosyrtes
Thanks for making this thread. Should be fun if it stays active for a bit...


Wait, to use TeX on TSR is it the exact same language used in the actual LaTeX program? Obviously without \usepackage and similar commands though.

Oh, and no problem :smile:
Original post by Elcor
Wait, to use TeX on TSR is it the exact same language used in the actual LaTeX program? Obviously without \usepackage and similar commands though.

Oh, and no problem :smile:


Yup. Enclose it in [ TEX] [ /TEX] (nospaces) and it works :smile:
Original post by Navo D.
A quick starter question that my physics teacher once gave me (don't google it - ruins the fun)

Show that ln(n!) = nln(n) - n + 1

There is a much more rigorous approach to this than what I used (but who knows just how smart some of you are) but it doesn't require anything beyond core modules.


Hang on, are you sure that's true?

ie if n=3

LHS=ln(6)=1.79
RHS=3ln(3)-3+1=ln(27)-2=1.29

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