Answer to “What’s wrong with this photo?”

31 December 2010

Original post here.

So, what’s wrong with the photo?

Answer: this is a trick question. It is not really a photo, it’s actually a painting!

http://www.drublair.com/comersus/store/tica.asp

An old (math) chestnut

31 December 2010

Theorem: $\sqrt[n]{2}$ is irrational for any integer $n \geq 3$.

Proof: Suppose the number is rational, and let $\sqrt[n]{2} = a/b$ where $a$ and $b$ are integers. Then $2 = (a/b)^n$, which can be written as

$b^n + b^n = a^n$.

This contradicts the Fermat’s Last Theorem. QED

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ADDENDUM: This is a perfectly correct proof, although it is akin to killing a mosquito using a bazooka.

Which reminds me of a story that happened 12 years ago. I was in Form 5 and had been called to attend the IMO selection camp at UPM. At the time, I knew much less mathematics compared to the other students, so I wasn’t so sure whether I could make the team.

During an introductory lecture, Prof. Abu Osman gave us this problem:

Given positive integers $a, b, c, n \geq 2$ such that $a^n + b^n = c^n$. Prove that $a > n$.

Being a wiseass, I immediately shouted out, “Solved it! Using the FLT we must have $n=2$ and we know that the smallest Pythagorean triple is $(3,4,5)$ so we’re done.”

One of the students in the camp quickly shot me down, and explained that we were not supposed to use any advanced results since we were doing high school contest problems. I protested, “but the FLT is true, isn’t it? Tanya prof kalau tak percaya.” He countered, “that is probably not the solution the author had in mind.” I then turned away and muttered in disgust, “who the hell cares what the author had in mind…” .

The intended solution goes like this: We can assume that $a \leq b$. From the equation, we have

$a^n = c^n - b^n = (c-b)(c^{n-1} + c^{n-2}b + \dotsb + b^{n-1})$.

Since $c>b\geq a$, we have $c-b \geq 1$, and the second bracket is $> na^{n-1}$. Therefore, $a^n > na^{n-1}$, and so $a>n$.

A math paper

30 December 2010

Today I came across this math paper:

Finding composite order ordinary elliptic curves using the Cocks-Pinch method, by D. Boneh, K. Rubin and A. Silverberg. (To appear in the Journal of Number Theory).

teehehe

Tigers vs bird

30 December 2010

Credit: popular-pics.com

What’s wrong with this photo?

29 December 2010

Look carefully:

Answer will be given after two days. Please do not use Google or Tineye, just use your own eyes to look at the photo. You can make the picture bigger by clicking on it.

Sudahkah anda

28 December 2010

…menjalankan tanggungjawab sebagai rakyat Malaysia? Mari mendaftar sebagai pengundi:

http://www.spr.gov.my/

Ahli Parlimen Batu:

27 December 2010