feb24.2008

Brain-tired from a gruesome four hour math contest, I am finally home, and sitting in front of my computer. The above image is one of Banksy's works (a British graffiti artist). I like his works, and his creativity. Check him out at:

Banksy

In any case, the math contest was another learning experience for me. Four hours. Five questions. I solved one - maybe two. I wasn't exactly surprised at it's level of difficulty, but I was disappointed that I wasn't able to fully solve a couple of them. For those who have any interest as to what the particular questions were:

1. There are ten cards with the number a on each, ten with b and then with c, where a, b and c are distinct. For every five cards, it is possible to add another five cards so that the sum of the numbers on these ten cards is 0. Prove that one of a, b or c equals 0.

2. Can it happen that lcm(1,2,...,n) = 2008 x lcm(1,2,...,m) for some positive integers m and n? (lcm(a,b,...,x) means the least common multiple of a,b,...,x.)

3. In triangle ABC, angle A=90º. M is the midpoint of BC and H is the foot of the altitude from A to BC. The line passing through M and perpendicular to AC meets the circumcircle of triangle AMC again at P. If BP intersects AH at K, prove that AK=HK.

4. No matter how two copies of a convex polygon are placed inside a square, they always have a common point. Prove that no matter how three copies of the same polygon are placed inside this square, they also have a common point.

5. We may permute the rows and may permute the columns of the table below. How many different tables can we generate?
1 2 3 4 5 6 7
7 1 2 3 4 5 6
6 7 1 2 3 4 5
5 6 7 1 2 3 4
4 5 6 7 1 2 3
3 4 5 6 7 1 2
2 3 4 5 6 7 1

And those are the questions.

In any case, I'll close with something new that I came across yesterday. Actually, Terry showed it to him, so it's courtesy of him. It's called Anaheim Machines, where a man by the name of Ivan Sanchez creates 3D models of various Gundams. It's incredibly, incredibly, ridiculous. Ridiculously amazing, that is.

Check out his work at:

Anaheim Machines