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# The Proof is Trivial! Watch

1. (Original post by Benjy100)
Problem 33

Determine
Solution 33

- upon expanding, completing the square and simplifying.

Now we could use a trig. substitution, but recognise that this is just a standard integral and so

since arcsin is an odd function.

2. (Original post by Lord of the Flies)
Solution 33

Method 1:

Letting gives:

Method 2:

(taking the principal branch - I'm sure that can be justified)
Too quick!
3. Solution 33

Letting we get

4. (Original post by aznkid66)
Lol, way to trip right in front of the finish line..

EDIT: Right, ignore this then.
You could've been a little nicer about it, even when you thought he was wrong

Do you see why und was right though?

Spoiler:
Show
5. (Original post by Lord of the Flies)
...
(Original post by Star-girl)
...
Both too quick ;_;
6. (Original post by und)
Problem 32

Find all positive integers such that and are both perfect squares.
By any chance is your next question going to be:

Spoiler:
Show

A triangle has sides of length at most 2, 3 and 4 respectively. Determine, with proof, the maximum possible area of the triangle.

7. (Original post by metaltron)
By any chance is your next question going to be:

Spoiler:
Show

A triangle has sides of length at most 2, 3 and 4 respectively. Determine, with proof, the maximum possible area of the triangle.

Nah, I wasn't planning to post that one.
8. Problem 34*

Prove that for all primes
9. Solution 34:

P is prime and greater than 3, so and must contain factors of 2, 3 and 4, so it is divisible by 24.
10. (Original post by Star-girl)
Problem 34*

Prove that for all primes

Since p>3, p cannot be divisible by 2 or 3. Hence, both p-1 and p+1 are even and one must be divisible by 4. Also, one of the three numbers, not p obviously, must be divisible by 3. Hence the whole expression is divisible by:

Edit : Dammit, I typed this so quickly as well!!
11. (Original post by metaltron)

Since p>3, p cannot be divisible by 2 or 3. Hence, both p-1 and p+1 are even and one must be divisible by 4. Also, one of the three numbers, not p obviously, must be divisible by 3. Hence the whole expression is divisible by:

Edit : Dammit, I typed this so quickly as well!!

(Let me have this one, I haven't got to any of the other ones first yet! )
12. (Original post by DJMayes)

(Let me have this one, I haven't got to any of the other ones first yet! )
Either have I Of course you can have it, you beat me to it!
13. I was just casually typing up the solution, but you guys...
14. (Original post by und)
I was just casually typing up the solution, but you guys...
... were not doing it casually.
15. ^^ I still haven't got to any one of them first.

Also nicely done, DJ and metaltron.
16. Problem 35*

Evaluate
17. Problem 36: */**

A particle is projected from the top of a plane inclined at an angle to the horizontal. It is projected down the plane. Prove that; if the particle is to attain it's maximum range, the angle of projection from the horizontal must satisfy:

18. (Original post by DJMayes)
Problem 36: */**

A particle is projected from the top of a plane inclined at an angle to the horizontal. It is projected down the plane. Prove that; if the particle is to attain it's maximum range, the angle of projection must satisfy:

Isn't this a STEP question you were discussing recently?
19. (Original post by und)
Isn't this a STEP question you were discussing recently?
No; there was one involving firing up an inclined plane but it wasn't this. I did go to put a question on this thread about proving the maximum range of a projectile occurred when the angle of projection was 45 degrees but figured that it was too easy so messed about a bit and came up with this.
20. (Original post by Lord of the Flies)
Problem 35*

Evaluate
Solution 35
I hope!
Solution 35

By parts we obtain

Again, by parts, we obtain

Let

Let

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