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# The Proof is Trivial! watch

1. (Original post by Jkn)
Problem 110*

Prove that for all real numbers with using Cauchy-Schwartz.
Solution 110

which one can verify is the same as , which is why the above quadratic is non-negative.

Since the equation is non-negative, this equation can have at most 1 root in (hopefully obvious) and this implies (discriminant test for no. of roots), from which the result follows.
2. (Original post by FireGarden)
Solution 110

Since all terms are non-negative, this equation can have at most 1 root in The this implies , from which the result follows.
Your result implies which is simply not true.

Also, why is it that all the coefficients being non-negative implies ? Consider etc... (perhaps i have misunderstood you. If so I apologise.)
3. (Original post by Jkn)
Your result implies which is simply not true.

Also, why is it that all the coefficients being non-negative implies ? Consider etc... (perhaps i have misunderstood you. If so I apologise.)
Wrote very badly/lazy.. to the point of making factual errors out of a lack of thinking.. but hopefully the corrections/additions should be sufficient now!
4. (Original post by FireGarden)
Wrote very badly/lazy.. to the point of making factual errors out of a lack of thinking.. but hopefully the corrections/additions should be sufficient now!
Well the infinite series I have presented are not necessarily bounded so are you able to justify the use of the discriminant in the quadratic equation and perhaps even the validity of forming a quadratic equation in the cases where the coefficients are infinite? (It may be correct, but you'll need to justify this assumption either way.)

Note that a very straightforward solution exists whereby the considerations of infinity are trivial (if it turns out your method is false.)
5. Subbing!
6. (Original post by elixir)
We have and for all .

How did you get these?
Can someone tell me how he got that implication, it's driving me nuts!

7. (Original post by elixir)
Can someone tell me how he got that implication, it's driving me nuts!

.
.

Solution 111

There are two possibilities:
If, , we have .
If, , then .

A quick question to Lord of the Flies:
Are you sure about problem 104? For example, the right-handed side is not defined when , whereas the left-handed side is.

(Original post by Jkn)
Hahahaa, did you figure out why?
Nope, I cannot think of anything that impels you to put such restrictions.
.
.

Solution 111

There are two possibilities:
If, , we have .
If, , then .

A quick question to Lord of the Flies:
Are you sure about problem 104? For example, the right-handed side is not defined when , whereas the left-handed side is.
thanks
Nope, I cannot think of anything that impels you to put such restrictions.
I found it when I looked at the Bulgarian Olympiads hahaha!

----------

Everyone who doing/is interested in STEP, check out the new thread I made: "STEP Mathematics Problem Solving Society"

I'm basically hoping to create something just like this but for STEP questions!
...
Typo, it is
11. Problem 112*** difficult..

Evaluate .

Problem 113*** I cannot resist posting number theory!

Let be a prime number and - positive integers. We have .

(Original post by Jkn)
I found it when I looked at the Bulgarian Olympiads hahaha!
It is given on the third round, right?
It is too easy for the fourth round; I have solved negligible number of questions of our third round.

(Original post by Lord of the Flies)
Typo, it is
Well. I am gonna ace it now.
It is given on the third round, right?
It is too easy for the fourth round; I have solved negligible number of questions of our third round.
That's right! Hahaaha! We can't all be Bulgarian geniuses . Come do some STEP on my thread! It's really empty
13. (Original post by Jkn)
We can't all be Bulgarian geniuses .
I know right
14. I was feeling smug about having solved someone's quantum mechanics problem. Then I returned here....
15. (Original post by bananarama2)
I know right
Fortunately though, we can be English geniuses
16. (Original post by Jkn)
Fortunately though, we can be English geniuses
You can! I can be hopeful.
17. (Original post by bananarama2)
You can! I can be hopeful.
We will work on it next year! Big up Emmanuel 2K13!
18. (Original post by Jkn)
We will work on it next year! Big up Emmanuel 2K13!
I think you and all the other freshers will think I'm some kind of social retard.
19. (Original post by bananarama2)
I think you and all the other freshers will think I'm some kind of social retard.
Hahahahahaha, **** it uni's a time to reinvent yourself

I will teach you some drinking games
20. Solution 112

Using standard formulae:

which upon integration with gives:

Note that

Which as . Hence we obtain the relation:

Hence:

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