This is amazing . This is something big .
I hope people will start to utilise this thread as much as possible . I'll share it to everyone I know .

Thank you so much .

Critical_Fallacy

I consulted the Bronze Medallist , and he told me that this solution is incorrect , and it'll be worth one point only . He attached a link to an accurate solution :

http://www.artofproblemsolving.com/Forum/v...ce9c11#p3152687

He added : The solution gives a construction of 2^k-1 terms but the problem statement requires exactly k terms

I never knew the questions in IMO would be this challenging !

This post has been edited by maximR: Aug 6 2013, 10:33 PM

I don't think I'll make the cut for IMO camps ( IPhO , probably . In the near future . ) though . You know the Bronze Medallist I mentioned ? He's as old as me , but he has far more experience and he started very young , at about 7 .

How do you find STPM ? Enjoying it ?

I see !

I've heard about turbulence in fluid motion and how hard it is to model that accurately ( I think it's still on Wikipedia's List of unsolved problems in Physics ) .

If you don't mind me asking , what led you to to become a Mathematical Physicist ? Was is a smooth pathway ?

I see . I've thought about that for a long time . Thanks for clarifying !

Now I've a question , it's regarding the constants that appear in Physics equations .

Example :

1 . E = mc^2
2 . F = kx
3 . F(grav) = G (Mm)/r^2

My question is , will a constant always appear if a physical quantity varies directly as the other ? If it will , why do some constants behave in different ways . Why is the speed of light squared in Einstein's equation ? Since there were no experimental data at first to support his theory , how did he derive c ?

Why can the constant in F = kma be defined so that F = ma , but not with other equations ?

I am aware of the actual equations .
But I've always wondered about the constants . Why must constants appear in every Physics equation ? And why can the constant in Newton's 2nd Law be simplified to 1 ?

Thank you for your reply .

What are you doing , currently ? Are you a Physics major ?

Thank you for your detailed explanation ! I really appreciate it .




Thanks !


This is what I need ! I've been thinking about this since Form Four . You've cleared everything up . Thank you !

As for why I proposed that there is a constant in F = ma , here's what's written in our textbooks , and reference books . One of the errors in our Physics syllabus . My aim is to major in Physics , then one day hopefully , revamp the Physics syllabus in Malaysia .

Here's the explanation provided by textbooks :

From Experiment 1 , a α F
From Experiment 2 , a α 1/m

The two results are combined .

a α F/m

OR

F α ma

Therefore : F = kma

Unit of force is newton , N .

In order to make the formula as simple as possible , we make k = 1 by defining a force of 1 N as

1 N is the force which gives a mass of 1 kg an acceleration of 1 m s^-2 ,

1 N = (k) (1 kg) (1 m s^-2)
k = 1

Therefore , F = ma

This post has been edited by maximR: Aug 12 2013, 04:22 PM

Another great post !
This is the first time I've read that Newton originally stated that Impulse is directly proportional to the Change in momentum . So he actually considered the time in which a force acts on a body . Interesting .

So if I get this right , the reason why G appears in Newton's Law of Universal Gravitation is because the unit of force is defined as amount needed to accelerate 1 kg of mass at the rate of 1 m/s² , therefore G should be added to ensure that the units on both sides of the equation are the same ? And that after careful experiments , experimental physicists discovered that the force on gravity between two bodies does not vary simply as F = Mm/r^2 , so there must be a constant to account for the interaction between the two bodies ?

And that means Newton never really had accurate calculations since he couldn't determine G , and did not include them in his equations ? ( or did he ? if yes , then how did he come to the conclusion that there is a constant just by empirical observation ? or did he derive it mathematically , or did he think that since the force of gravity is usually very small between two bodies , there must be something that accounts for this , therefore , a constant must exist ? )



This post has been edited by maximR: Aug 13 2013, 01:19 PM

I was well aware that Newton did not explicitly state that F = ma ( It's derived from his 2nd law ) , and I've read that force is defined as the rate of change of momentum early on when I first started learning dynamics . Did you really not read anything other than your SPM Physics reference book in form four and form five ?

But I didn't know that he said Ft α mv - mu

This post has been edited by maximR: Aug 13 2013, 10:49 PM

For beginners like me , how do I develop the intuition to manipulate the equation so p exists in that KE equation ?
Also , if KE becomes 2KE , p becomes 2p as well ? How to prove this ?



All right , noted . Thank you .

Here's a question from an SPM Olympiad Fizik book :



This post has been edited by maximR: Aug 14 2013, 06:24 PM

According to the guy who provided the question , yes .

Solution provided is about pressure , artificial gravity , inertia ...

I get it now .

But looking for explanations on the internet , there are some who use General Relativity to explain this . So is the explanation using air compression / buoyant force correct , or General Relativity , or both ?

All right . Thank you ! I got only some of them covered through watching Walter Lewin's lectures , like proving the Work-Energy Theorem , P= h(rho)g , etc . I learned the general proof for derivatives too , but not the SPM way where you just add delta x and delta y to x and y respectively . I prefer the usual method , those that are available online .

Sorry I went offline yesterday , it was quite late and I had to go to bed . Anyway , I'm done with Remainder Theorem and Factor Theorem .

As for your last question , is my reasoning correct ?

Since the remainder theorem states that if f(x) is divided by (x-a) , then the remainder = f(a) and factor theorem states that if f(x) is divided by (x-a) and it equals zero , then (x-a) is a linear factor , thus , if x = a , then :

a^3 - 6a - 7a + 60 = 0

Factoring using my calculator , since I haven't learned Cubic and Quartic Eqn ( can you please re-post the tutorial ? it's too small for my eyes ) yields :

(a+3)(a-5)(a-4) = 0

Therefore :

(x+3)(x-5)(x-4) = 0

Thus , I've found the linear factors of the function without using the first linear factor . I hope it's correct .

Actually , I wanted to tell you that the day after you posted that Cubic/Quartic tutorial , I'd read it and it's safe to say I'm done with it . Maybe more practices so I can be faster with algebraic long division .

I'll look through these tutorials probably after my Biology paper . Thank you so much , someone who spends his time creating tutorials for a complete stranger deserves an award . Don't really know how to re-pay you .

I'm back , after a couple of days of rest .
Critical_Fallacy there has been quite a lot of things going on here , and I'll take some time to slowly wade through . I think I'll start with where I left off .

RED-HAIR-SHANKS may I know where you get these questions ?

A quick question as I have been reading a book on core concepts in Maths , the chapter is Introduction to Logic .

" Can a statement such as 'Some men are silly' be disproved by finding an example of a man who is not silly ? Discuss what type of statement we should have to have in order to be able to disprove it by finding a counterexample .'

I was thinking along the lines of 'All men are not silly' , then coming with 'This man is silly' which is a counterexample of the negation , but that would he proving that initial statement .

By the way , I see a lot of new tutorials ! Will update about my progress tomorrow .

Thank you . I've been placing my hand in different cookie jars , not knowing which to delve deeper into . For now , I have the luxury to do so so I'll jump around different topics , and get some serious work done in January .

The problem which I posed is from the book titled ' Basic Concepts of Mathematics and Logic ' by Michael C . Gemignani , which covers logic , set theory , counting , numbers , functions , ordering , probabilities and Euclidean geometry . It's quite a compact book , not the best designed book but I find it quite entertaining ( although the content comprises mostly of paragraphs of words ) .

That problem comes after this :

In order to disprove the following statement , which statements would we actually prove ?

' Some triangles are isosceles . '

The answer is : All triangles are not isosceles . (the answer doesn't delve deeper into how to prove this statement)

After the topic on Mathematical Disproofs , I ventured into Conjunctions and Disjunctions , which I've seen when learning Mathematical Reasoning in SPM Mathematics , and Boolean Algebra in SPM Physics .

A question : If A , then B is equivalent to ~( A ∧ ~B ) , with the reasoning :

If A , then B , thus if A occurs B must occur as well , or it cannot happen that A occurs but B does not occur , it cannot happen that A occurs and ~B occurs .

So this statement ~( ~A ∧ B ) is false , right ? Because if A does not occur , B can occur .

At the Big Bad Wolf sale , I bought Calculus ( Pearson International Edition ) textbook for RM 10 , which is a huge college level textbook which covers a lot of concepts in Calculus . I'm still in the preliminaries section , I'm familiar with most of the topics in the preliminaries ( they are covered in SPM Maths / Add Maths ) so I'm learning Logic and Proofs . I've also printed out your tutorials , but not sure where to start with all the resources around me .

In the Calculus textbook , there are a few problems which I need help with ( college level textbooks only provide solutions for Odd-Number questions )

1 . We can show that a number divided by zero is meaningless . Suppose a ≠ 0 . If a/0 = b , then a = 0 , which is a contradiction . Now find a reason why 0/0 is also meaningless .

2 . Show that any rational number p/q , for which the prime factorisation of q consists entirely of 2s and 5s , has a terminating decimal expansion .

3 . Show that between two real numbers there is a rational number .

4 . (a) Use the Fundamental Theorem of Arithmetic to show that the square of any natural number greater than 1 can be written as the product of primes in a unique way , except for the order of the factors , with each prime occurring an even number of times .

(b) Show that √2 is irrational . ( Hint : Try a proof by contradiction . Suppose that √2 = p/q , where p and q are natural ( necessarily different from 1 ) . Then 2 = p^2/q^2 , and 2q^2 = p^2 . Now use (a) to get the contradiction .

Your help is greatly appreciated . By the way , is this topic under Number Theory ? It's under preliminaries , the author didn't label this sub-topic .

A very good exercise as I'm studying ways of proofs and logic , I haven't abandoned all Maths stuff yet .

Anyway , if this were a simple irrational X irrational problem , it would be easier to come up with a proof . The exponent makes it a little bit more complicated for me . I tried hard , but gave up in the end . So I Googled for it , and I can say that I'm 'mind-blown' !

I need help for the proof for ' rational between two reals problem ' . The book I have gives hints about the Law of Completeness and something about set theory which I can't comprehend .