Solve it in ten minutes!

My friend heard a problem from his friend:

Find a positive integer solution to 4024x + 4178y + 4609z = 3757057.

Yes, this is a not-so-interesting question if you have a computer and you know how to write a programme (just do some brute-force calculation).

Also, if you requires an integer solution (so you may have non-positive integers as x, y or z), it is still easy to do it by hand quickly. (Do you know how?)

However, assume what you have is a sheet of paper and a pen, can you find a positive solution in ten minutes?

三角題一則

問題：求 3 sin x + 4 cos x 的最大值。

在會考的附加數學科裏這題有以下的標準解法：設 3 sin x + 4 cos x = R sin (x+y)，則 3 sin x + 4 cos x = (R cos y) sin x + (R sin y) cos x，故比較系數可知 R cos y = 3 而 R sin y = 4（這時頭腦清醒的學生應該問「為何可以比較系數？」），解方程可得 R = 5 而 y = tan^-1(4/3)，從而 3 sin x + 4 cos x 的最大值是 5。

最近碰到以下有趣的題解：不妨設 x 為銳角（為何可以這樣做？），再考慮下圖，易見 CE = 3 sin x 而 ED = 4 cos x。再者，不難發現 AEB 是直角而 AC // BD，故此 3 sin x + 4 cos x = CD ≦ AB = 5（等號何時成立？）。

香港大學公開講座

日期：2008 年 12 月 6 日（星期六）
時間：下午 2 時 30 分至 4 時
地點：香港大學許磐卿講堂（LE1）
講者：吳端偉博士
講題：拍賣中尋對策

有關其他詳情及查詢電話，可瀏覽 http://147.8.101.93/math/2008dec/2008dec06.pdf。

Analytic function on a disk

Suppose f is a holomorphic function on the unit disk that is continuous up to the boundary. If f vanishes on an arc of the boundary circle, show that it is identically zero.

This is a standard question from complex analysis. I just learn today that there is a really cute solution to it. You may want to think about it before continuing.

The classical method is to use Schwartz reflection principle and argue that you can analytically continue the function outside the disk a little bit; then the continued function vanishes on a segment, so it must vanish identically.

The cute solution that I was referring to is the following: take (finitely many) copies of f, rotate each of them suitably and multiply the rotated functions altogether. Then the product is going to vanish identically on the boundary of the unit disk, and of course the product is holomorphic inside the disk. Hence the product is identically zero, and thus f is identically zero. (Just argue that all derivatives of f vanishes at the origin by differentiating the product.)

Factorial

Today I went to a talk by Manjul Bhargava, and he stated some interesting facts in number theory that has to do with the factorial:

1. If are integers, then is a multiple of .

2. Suppose that f is a primitive polynomial with integer coefficients and let k be its degree. Let Let d(f) be the gcd of all f(a) as a runs through all the integers. Then d(f) divides k!.

3. The number of polynomial maps is .

4. A function is continuous if and only if it has the form where as .

In fact in the talk he gave a far reaching generalization of the factorial functions. For each compact subset of the p-adic rationals (e.g. the p-adic integers), he defined a factorial function adapted to that set such that the above seemingly unrelated facts about the factorial goes through. It's amazing to see how he could generalize things that are so well known, and give non-trivial results that fits in so many settings.

[I just realize that his original article can be found here.]