Maths Tournament

[Charles Pegge]

A new member of our forum, a young mathematician posted this link to me. A wide range of mathematical problems are posed. Would anyone like to have a go at some of them?

http://www.itym.org/

C (http://www.itym.org/)harles

[danbaron]

(I don't have any confidence, but, I think the best way to stretch your brain and learn, is to not be too afraid to be wrong.)

ITYM2009

1. Specular Colourings

1. What is the maximum number of cells of an m × n grid that can be coloured blue, such
that no two blue cells are symmetric with respect to any horizontal or vertical line of the
grid?

If m * n is even, then, m * n / 2.

If m * n is odd, then, (m * n - 1) / 2 + 1.

[Charles Pegge]

Another problem we discussed at some length is how to turn a sphere inside out without breaking it's surface. :)

Charles

[danbaron]

I think for that one you need 4 spatial dimensions.

If you can create such an environment, I would be very interested.

[danbaron]

2009ITYM

8. Positivity of Symmetric Polynomials

A polynomial P(x, y) with real coefficients is symmetric if the equality P(x, y) = P(y, x)
holds for all x, y in R.

1. Let P(x, y) = x^3 + ax^2y + axy^2 + y^3 be a symmetric polynomial of degree 3. Prove that
P(x, y) > 0 for all x, y > 0 if and only if a > -1.

(I wish we had math notation.

The problems don't even display correctly.

And, using computer notation in the solutions is unpleasant.

But, it's no one's fault.)

Anyway, I think this solution is in the neighborhood of being correct, but, I bet there are easier and more precise ways to do it.

f(x,y) = x^3 + ax^2y + axy^2 + y^3

f(x,y) = x^3 + y^3 + axy^2 + ax^2y

Let, y = x/c.

f(c) = x^3 + x^3/c^3 + ax^3/c^2 + ax^3/c

f'(c) = - 3x^3/c^4 - 2ax^3/c^3 - ax^3/c^2

We want to find the extreme value of f(c), so, set f'(c) = 0.

Then,

- 3x^3/c^4 = a( 2x^3/c^3 + x^3/c^2 ) [1]

Equation [1] is satisfied only if c = 1, and a = -1.

So, when c = 1, f(c) is an extreme.

But, is it a maximum or a minimum?

We've already determined, a = -1.

Then,

f(c) = x^3 + x^3/c^3 - x^3/c^2 - x^3/c

We know that, f(1) = 0.

f(1/2) = x^3 + 8x^3 - 4x^3 - 2x^3 = 3x^3

So, f(1), is a minimum.

f(c) takes the minimum value of 0 when c = 1, and, a = -1.

For positive f(x,y), this implies a > -1.

[ErosOlmi]

http://www.thinbasic.com/community/showthread.php?9826-TBGL-Whatsit&highlight=Quaternion

[John Spikowski]

Here is a PDF (http://sites.google.com/site/wwwitymorg/problems/Problems-ITYM2011.pdf?attredirects=0) of the problems for this years competition.

I need to get back to work on the GSL (http://www.gnu.org/s/gsl/) extension module for ScriptBasic.


Another problem we discussed at some length is how to turn a sphere inside out without breaking it's surface.

Here is my warped view of the world (http://files.allbasic.info/AllBasic/bluemarble.html).