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# The Proof is Trivial! Watch

1. Solution 117

Let be a solution to .

Since and we have . Set gives and hence since cont.

It is now clear that if are odd and otherwise. If then we require , if it is easily checked that has three possible values when is even and two when is odd both these cases include of course

One of my favourite series questions:

Problem 119***

2. Solution 119

We use generating functions.

Observe
Hence
Thus

.
Further, , which is equivalent to .
Linear differential equation, with variable coefficients. We employ Lagrange's method. After solving this equation (it is too late here, so I am omitting the details (I may add these details in tomorrow)), we obtain .
Solution 119
Bravo! I took a somewhat different route, if you're interested.

which is very easy.
4. Oh and here is a quick alternative solution to 108:

Solution 108

are integer roots to the eq. exactly one of is odd/even. Let be the odd root:

LHS is odd, RHS is even, hence cannot exist. So the polynomial is not a quadratic
5. (Original post by Lord of the Flies)
I took a somewhat different route, if you're interested
Superb. I would have never come up with this solution, for when I saw the binomial coefficient, I instantly started thinking of combinatorial solution.

The following problem is inspired by your profile picture.

Problem 120***

Evaluate: , where .
6. Solution 120

Therefore
7. (Original post by Lord of the Flies)
Solution 120
You would agree that Laplace is indeed quite useful for this problem, would you not?
I suspect that there is a complex analysis solution, but I am not sure if it is rather laborious.

Problem 121*** It is not that awful.

Evaluate

Problem 122*** This is tough, unless one has studied analytic number theory.

Let , and . Find an explicit form for .
8. Problem 123***

Prove that (Mod p) is solvable for every prime p.
9. Solution 123

For the problem is trivial. We suppose . Note that is solvable if and only if . To prove this assertion, we introduce - primitive root . Let and . We then have . This is equivalent to . Let . If then is solvable and . If , then is not solvable and .
Hence we have to prove that , which is obvious unless . However, in this case ; thus is solvable and we are done.
10. ...alright then.

Problem 124

Evaluate

Where is the legendre symbol.
11. Some of these questions are getting a little silly (for the intended audience). Laplace transforms? Legendre symbols? These are things encountered in the second or third year of a good university. Having said that, if people are happy and able to solve them, by all means be my guest
12. Solution 121

In retrospect I am not sure switching to polar was the best idea, since some of the working is a bit tedious, but whatever.
13. Solution 124

Denote . Hence we have . Let be such that . Hence

Therefore

(Original post by Lord of the Flies)
Solution 121
In retrospect I am not sure switching to polar was the best idea, since some of the working is a bit tedious, but whatever.
This integral turned out to be an annoying computation (I have also solved it using polar coordinates). I regret posting it, since most probably there is no concise solution.

Problem 125**

Let be students. Some of these students know each other. Show that we can split these students into two groups such that if given student knows students from his or her group, then he or she knows at least students from the other group.

Problem 126***

Let be a continuous non-decreasing function. Show that .
14. This isn't particularly fair, but whatever here goes (they're pretty ):

Plot

Plot
15. (Original post by shamika)
This isn't particularly fair, but whatever here goes (they're pretty ):

Plot

Plot
Well, according to Wolfram Alpha: and
16. (Original post by Zephyr1011)
Well, according to Wolfram Alpha: and
Yep

The first is called the butterfly curve - the second is my own creation after playing around with it a bit
17. (Original post by shamika)
Yep

The first is called the butterfly curve - the second is my own creation after playing around with it a bit
What do you call this then?
Spoiler:
Show

Let

and the functions

Plot and on the same graph.

Spoiler:
Show

Alternatively, if parametric graphs are more your cup of tea:

Let

Plot where

Spoiler:
Show

You can tell I'm bored.

18. (Original post by ukdragon37)
What do you call this then?
Spoiler:
Show

Let

and the functions

Plot and on the same graph.

Spoiler:
Show

Alternatively, if parametric graphs are more your cup of tea:

Let

Plot where

Spoiler:
Show

You can tell I'm bored.

LOL! At least you could stick my curve into Wolfram Alpha easily (which is what I was intending )

What is it?
19. (Original post by shamika)
Some of these questions are getting a little silly (for the intended audience). Laplace transforms? Legendre symbols? These are things encountered in the second or third year of a good university. Having said that, if people are happy and able to solve them, by all means be my guest
I find it telling how the OP hasn't submitted a solution since about question 50, and we're now past question 120. This thread stopped being interesting (Or even accessible) for the vast majority of people quite a long time ago and now it seems very much to just be a battle of wits between Mladenov and Lord of the Flies.

(If this comes across as jealous, then that's probably because it is - I'd love to be able to even attempt the questions that are being thrown around lately! Unfortunately, the questions that are being thrown out are aimed at a very small audience indeed and just looking at the list in the OP it's very easy to see. It reminds me very strongly of what the STEP thread was like before Christmas.)
20. (Original post by DJMayes)
I find it telling how the OP hasn't submitted a solution since about question 50, and we're now past question 120. This thread stopped being interesting (Or even accessible) for the vast majority of people quite a long time ago and now it seems very much to just be a battle of wits between Mladenov and Lord of the Flies.

(If this comes across as jealous, then that's probably because it is - I'd love to be able to even attempt the questions that are being thrown around lately! Unfortunately, the questions that are being thrown out are aimed at a very small audience indeed and just looking at the list in the OP it's very easy to see. It reminds me very strongly of what the STEP thread was like before Christmas.)
France vs. Bulgaria.

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