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# The Proof is Trivial! Watch

1. Problem 509

Find
2. Solution 509

Let
Therefore

Subtracting ** from *:

3. (Original post by 16Characters....)
Solution 509
Very nice!

There's a neat trick to it at as well,
Spoiler:
Show
Differentiate the geometric sum formula
4. (Original post by Zacken)
Very nice!

There's a neat trick to it at as well,
Spoiler:
Show
Differentiate the geometric sum formula
Nice. I'll finish my current graphs question and then try and think up some not involving matrices.
5. (Original post by 16Characters....)
Nice. I'll finish my current graphs question and then try and think up some not involving matrices.
I need to go practice some STEP, III 2000 is calling out to me.
6. (Original post by Renzhi10122)
Problem 509

Let be the sides of a triangle. Prove that
Spoiler:
Show
This is equivalent to (eg via algebra bash ie expand everything out)

which is true by the triangle inequality, assuming non-degeneracy.
7. (Original post by metaltron)
Spoiler:
Show
This is equivalent to (eg via algebra bash ie expand everything out)

which is true by the triangle inequality, assuming non-degeneracy.
Yep, quite nice I think.
8. (Original post by Zacken)
Problem 503

Given integers and satisfying , find integers and such that

Solution 503

Multiply out by bd.

We may assume that so . Likewise, so . 2 divides exactly one of b-a and c+d. Thus, we get two solution for c and d:
and
and these do indeed work
9. (Original post by Renzhi10122)
Solution 503

Multiply out by bd.

We may assume that so . Likewise, so . 2 divides exactly one of b-a and c+d. Thus, we get two solution for c and d:
and
and these do indeed work
I agree with you up to the before last line, love the solution although it was a little overkill!

If we have

So, and definitely works, I agree.

But
10. (Original post by Zacken)
I agree with you up to the before last line, love the solution although it was a little overkill!

If we have

So, and definitely works, I agree.

But
Very good point, ignore the last line
11. Problem 510*

Let be a point inside a triangle . Prove that:

12. (Original post by Renzhi10122)
Problem 510*

Let be a point inside a triangle. Prove that:

Presumaby A, B, C are the vertices of the triangle?
13. (Original post by 16Characters....)
Presumaby A, B, C are the vertices of the triangle?
Woops, yes they are, let me edit that...
14. (Original post by Zacken)
Problem 509

Find

Therefore

Set we have the required sum as .
15. (Original post by Kummer)
x
Nice! This is a pretty involved solution, you're pretty good at this?

Are you at uni?
16. (Original post by Zacken)
Nice! This is a pretty involved solution, you're pretty good at this?

Are you at uni?
Not that anyone would need this technique for this sum, but I think different solutions are good. Yes, I'm at uni.
17. (Original post by Zacken)
Problem 509

Find
Seems like a binomial expansion will work here:

It follows that
18. (Original post by Renzhi10122)
Seems like a binomial expansion will work here:

It follows that
I think you can just look at Sn and 0.5Sn, take one from the other, and be left with a geometric series... though I haven't put pen to paper with this one.

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19. (Original post by Krollo)
I think you can just look at Sn and 0.5Sn, take one from the other, and be left with a geometric series... though I haven't put pen to paper with this one.

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Yep, I saw, just thought I'd give my solution.
20. (Original post by Renzhi10122)
Yep, I saw, just thought I'd give my solution.
Good good. Always nice to have a good variety

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