Cool problem of the day! (1 Viewer)

[nightweaver066]

Very nice Realise!

nightweaver, you are very close, but not quite.

A few things:

1. It's called the Triangle Inequality, not the 'Axiom of Triangle'. It is something prove-able.

2. 2 is indeed a lower bound for x, but it isn't necessarily the LARGEST lower bound. Consider the limiting cases when the triangle is right angled.

Updated.

 

[Bored_of_HSC]

Updated.

Haha you got 'axiom of triangle' from Harry didn't you?

 

[nightweaver066]

Haha you got 'axiom of triangle' from Harry didn't you?

Who else.. haha..

2 root 7 instead of 4 root 2.

I can't do simple arithmetic. Thank you.

 

[Carrotsticks]

Who else.. haha..

I think best to use 'Triangle Inequality' from now on.

'Axiom of triangle' just sounds silly imo as it is not an axiom.

 

[barbernator]

sorry I phrased the inequality completely wrongly. I don't think it's really an inequality I have to solve as an upper bound of seaniegs question now i think of it. The lower bound was just x^2 + y^2 +1 > 2xy, but i've got no idea for how to prove that z<4. It's probably something to do with the discriminant being a perfect square when i solve the quadratic for x possibly, but I don't really know how to prove that anyway. Oh well.

 

[Demento1]

If only I was capable enough to do the questions which Carrotsticks posts up daily. I must be missing out.

 

[barbernator]

If only I was capable enough to do the questions which Carrotsticks posts up daily. I must be missing out.

in time you will btw, i have a good one which i will suggest to carrot for the next round of q's

 

[seanieg89]

sorry I phrased the inequality completely wrongly. I don't think it's really an inequality I have to solve as an upper bound of seaniegs question now i think of it. The lower bound was just x^2 + y^2 +1 > 2xy, but i've got no idea for how to prove that z<4. It's probably something to do with the discriminant being a perfect square when i solve the quadratic for x possibly, but I don't really know how to prove that anyway. Oh well.

Proving that z>2 is trivial, but proving z<4 is not...Assume solutions with z>3 exists and look for a contradiction. I will post a solution tomorrow.

 

[Carrotsticks]

Hint for the Triangular Number question:

First prove that the sum of two consecutive triangular numbers is a perfect square....

 

[seanieg89]

 

[Carrotsticks]

Sweet!

Here is the next set of problems:

Geometry (courtesy of barbernator):

Consider a circular paddock with radius R such that R > 5 metres.

Along the circumference of the paddock, a post hammered into the ground and a sheep (poor sheep) is tied to the post by a 10 metre long rope.

In terms of R, what percentage of the paddock can the sheep roam?

Note: To check if your answer is correct or not, you should be able to show that when R --> Infinity, the percentage approaches 0 and when R --> 5, the percentage approaches 100.

Algebra:

Prove that the sum of the reciprocals of INFINITE Triangular Numbers is 2.

Number Theory:

seanieg89 will be providing the Number Theory questions from now on. Much appreciated =)

 

[RealiseNothing]

Algebra:

Prove that the sum of the reciprocals of INFINITE Triangular Numbers is 2.

I've noticed that the triangular numbers are either multiples of 3, or you can make a series of the non-multiples of 3 which differ by increasing factors of 9.

This have anything to do with it?

 

[seanieg89]

New number theory questions, ordered roughly by increasing difficulty:

1. Prove that the product of 5 consecutive positive integers cannot be the square of a positive integer.

2. The positive integers m and n satisfy: . Prove that m-n is a perfect square.

3. An odd positive integer n is said to be 'sexy' if n divides: Prove that for any twin prime pair (p,q), EXACTLY one of p,q is sexy.

4. Let a be an arbitrary irrational number. Prove that there are irrational numbers b and b' such that:

-a+b, ab' are both rational,

-a+b', ab are both irrational.

 

[qwerty44]

I never saw the day coming where you would prove something is sexy...

 

[seanieg89]

And a quick sketch of my solution to the question I posted earlier, I might tidy it up a bit later.View attachment amstemplate.pdf

 

[barbernator]

ahhh i see

 

[Sy123]

Please revive this!
I would love more questions to come along, some of them albeit hard are fun to do, and it excercise obscure math techniques.

 

[AAEldar]

Please revive this!
I would love more questions to come along, some of them albeit hard are fun to do, and it excercise obscure math techniques.

I've got an idea for one, just have to go about typing it up >.< I'll try and have it up by tomorrow - Carrot will hopefully have his bunch up as well, but he is a busy man!

 

[Carrotsticks]

Please revive this!
I would love more questions to come along, some of them albeit hard are fun to do, and it excercise obscure math techniques.

Sorry, been a bit busy lately. I might be able to do something tonight.

I've got an idea for one, just have to go about typing it up >.< I'll try and have it up by tomorrow - Carrot will hopefully have his bunch up as well, but he is a busy man!

That would be great! Please PM me =)

 

[Carrotsticks]

Geometry:

Consider a circular ring of radius R.

A stick of length k such that is placed within this circle so it becomes a 'chord'.

The stick is then 'spun' around inside the circle such that the two ends of the stick are always touching the circumference of the circle.

Find the area of the shape created by the midpoint of the stick once it's made a full rotation.

Algebra:

Determine all real values of X and Y satisfying the following system of equations:



University students/Postgrads: Please refrain from answering, as tempting as it may be.