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# The Proof is Trivial! Watch

1. (Original post by Lord of the Flies)
I had already solved it but was eager to post a solution to 98, after that blunder. Also, it doesn't matter what order the solutions come in - this thread is not a competition, it's for fun!
Haha I swear you never make mistakes!

I know, I know Tbf fair I only really typed it to practice my latex
2. (Original post by Jkn)
France vs. Bulgaria
I would genuinely pay to see a maths-off between these two Stick L'art in there as well
3. Solution 105

I shall not consider convergence; if somebody is interested, I can justify my approach.

Let us see..

From Cauchy, we have
.
Hence,
.

Edit: I was to post a solution to problem 108, but just saw the spoiler.
4. (Original post by Felix Felicis)
I would genuinely pay to see a maths-off between these two Stick L'art in there as well
Me too "Student room international Olympiad". As long as I get to set the questions I'm happy

(Original post by Mladenov)
Edit: I was to post a solution to problem 108, but just saw the spoiler.
Hahahaa, did you figure out why?
5. (Original post by metaltron)
I got 180 points I think for a Number Theory Question, it was Q8 but it was similar to STEP I Q1 2000.

For Combinatorics I got 180 points too, but also got three wrong answers for one so that ruined the moment for me really.

I think you can only get 180 pts for a question in Level 3.
Oh yeah that one was quite funny, I took seconds! That means theres a 6/8 overlap between levels (thats level 4's 3rd hardest one atm!)

Awesome man. Yeah level 4 only goes up to 230
6. (Original post by Mladenov)
Splendid solution.
You are not using continuity, so it is not necessary.

Another approach:
Spoiler:
Show
Clearly, is a solution. Suppose that there is at least one such that . We have and for all . Since is bounded below and satisfies , it follows that , . Note that implies . Hence all the solutions are and .
We have and for all .

How did you get these?
7. Since there is some love for function based questions..

Problem 109*/**

Find exactly
8. (Original post by FireGarden)
Since there is some love for function based questions..

Problem 109*/**

Find exactly
Not sure if this is right...

Assume x is not a function of t.

Letting x=t gives .

So
9. (Original post by Jkn)
Not sure if this is right...

Solution 109

Assume x is not a function of t.

Letting x=t gives .

So
t is simply a dummy variable for the integral, it doesn't really need to be considered all that much.

Edit: Realising that it wasn't previously clear at all, it's not correct.
10. (Original post by FireGarden)
t is simply a dummy variable for the integral, it doesn't really need to be considered all that much.
Is it right then?

Well then how can you define a dummy variable? And why is it you you can assume that x can be treated as a constant in the integral?
11. (Original post by Jkn)
Is it right then?

Well then how can you define a dummy variable? And why is it you you can assume that x can be treated as a constant in the integral?
I edited my previous post.

x is certainly a constant in the integral; giving the function f a value of x defines the limits and the x inside the integral as a constant.
12. (Original post by FireGarden)
Since there is some love for function based questions..

Problem 109*/**

Find exactly
Partial solution (will finish if I get time later).

Spoiler:
Show
Differentiate both sides to get f'(x)=cos(x)+xf(0)=cos(x)+cx

so f(x)=sin(x)+cx^2/2+d for constants c and d. By plugging this into the original equation we may get conditions that c,d must satisfy.
13. (Original post by james22)
Partial solution (will finish if I get time later).

Spoiler:
Show
Differentiate both sides to get f'(x)=cos(x)+xf(0)=cos(x)+cx

so f(x)=sin(x)+cx^2/2+d for constants c and d. By plugging this into the original equation we may get conditions that c,d must satisfy.
You have carried out your method incorrectly. (Though it was the method I was expecting)
14. (Original post by FireGarden)
I edited my previous post.

x is certainly a constant in the integral; giving the function f a value of x defines the limits and the x inside the integral as a constant.
Well then if it is a constant then why is it denoted most the most universally recognisable symbol for a variable?

And if it is a constant then its derivative is not f'(x) so how is it that you would expect this?
15. Solution 109

Hence

16. (Original post by Jkn)
Well then if it is a constant then why is it denoted most the most universally recognisable symbol for a variable?

And if it is a constant then its derivative is not f'(x) so how is it that you would expect this?
Do you believe to be an ill-defined function?

Take some particular value, , and you get

My function works similarly, and is just as well-defined.
17. (Original post by Lord of the Flies)
Solution 109

Hence

Sign mistake! Wait!
Correct. I was about to question your sinh(x), but saw your edit as I quoted you!
18. (Original post by FireGarden)
Correct. I was about to question your sinh(x), but saw your edit as I quoted you!
Yes I started with the wrong equation

(Original post by Felix Felicis)
...
That sig.
19. (Original post by FireGarden)
Do you believe to be an ill-defined function?

Take some particular value, , and you get

My function works similarly, and is just as well-defined.
Hmm I suppose I'll just ponder it some more (never really since this type of function before). Thanks
20. Problem 110*

Prove that for all real numbers with using Cauchy-Schwartz.

Problem 111* (another problem I've adapted from something else. Oh how I love generalisations...)

Let a,b and be positive integers such that . Find the minimum value of for the different cases that arise according to the value of given that has 4 proper factors (i.e. factors not including itself and 1.)

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