The International Mathematical Olympiad 2006

Problem 1. Let ABC be a triangle with incentre I. A point P in the interior of the triangle satifies:

Show that , and that equality holds if and only if .

A summary from the maths-learning group

Right now I am very pleased for where my maths-learning group is going. In fact, I am very confident that I can finish my plan written several days ago. ( maybe with some changes.) I try to make small sub-plans for no more than ten days a time. These small plans are very specific and easy to realize. In this way I can do the work more efficiently and better.

Three proofs of a identity

In this post, I will give three proofs for a important identity, which all come from Stein’s book:’real analysis’(both in the context and in the exercises):

The first proof:

The first proof has more of a ‘complex analysis’ taste, since it uses the properties of singularities of an entire function.

We first prove the following

First we observe both the left-hand side and the right-hand side have the following three properties (stated as ):

Entire functions

I have just finished the first read of Chapter 5 of Stein’s book: complex analysis. Roughly put, entire functions are decided by there zeros,only with difference of a multiplier, which is a entire function at the whole complex plane without zeros. It can also be represented as the form , where is entire. If the number of zeros are finite, then the function can be constructed as followings

But when it comes to the situation when the number of zeros are infinite, it is much more complicated. However, we are lucky enough to have the following theorem:

Theorem 4.1(weierstrass infinite products) Given any sequence of complex numbers with as , there exists an entire function f that vanishes at all and nowhere else. Any other such entire function is of the form , where g is entire.

To prove this theorem, we should observe the important identity first:

There are lots of ways to prove this identity, which I will write another post to make a summary. Observe that the zeros of the right hand side are precisely all the integers.

Basically what we will do to construct our entire function is to make our act just like these integers. A problem we should pay attention to is that the speed with which grows. Till now we can see, some very detailed analysis must be done in order to get this problem solved, and that is what the whole story goes. So, there are lots of paragraphs dealing with the growing speed. The first section, namely Jensen’s formula is a devise to do this. With the help of Theorem 2.1, Hadamard’s factorization theorem goes even further, strengthening the weierstrass infinite products.

(updated on Nov.10: ‘only with difference of a multiplier, which is an entire function at the whole complex plane without zeros. It can also be represented as the form , where is entire.’added, adviced by Akhil Mathew)

A plan for maths studying in the next a few months

This is basically the same post I wrote several days ago in Chinese, which is a plan to learn maths in the next a few months. I hope more people can join me so I rewrite it in English. There are so many things that I want to learn. So the following plan looks a little challenging. I will only take two materials at the same time, and I haven’t decided the sequence of them. I also set up a google group for this. Anyone who is interested is welcomed to join it at

http://groups.google.com/group/maths-learning

A comment to one post from Professer Laba

The post ‘paint it red’ was written by Professor Laba on her blog.

Dear Professor Laba:

I never know you came from Poland, a country which looks just like China in so many ways. Your article reminds me some of my memories, and the stories that I heard from my parents. Some say China is already a better place and I can live with that argument, though I am not satisfied with the current situation. Yes, it is true that we generally don’t get punished to death simply because we have said or write something wrong, it is true we don’t send all the intellectuals into the country to receive ‘reeducation as a farmer’ simply because the party won the ‘revolution war’ relying on farmers. But, people’s minds are devastated, with little free will left. They already get used to the giving up the opportunity of thinking by themselves. They don’t even try any more. Most only follow orders of their ‘the big leaders’. So most social problems existing in China today can track back to several decades ago, when ‘FREEDOM IS SLAVERY’, and ‘IGNORANCE IS STRENGTH.’ (George Orwell nineteen eighty four)

十一月至明年上半年的学习计划

我一年之前还常常做计划，然后发现总是不能完成。原因是候我总是有自虐倾向。以往做计划存在两个比较大的问题，其一是没有把计划内容做到切实可行，选了过多过难的材料；另一个问题是只有比较长时间的计划，而没有短期到一周甚至一天两天的具体安排。当然，做出这样的安排也是很困难的事情，因为学习不比做工程。常常事先很难断定完成一个任务所需要花费的精力。数学的学习更是如此。

所以我需要的一个灵活的计划。能够及时的调整先后顺序，而且易于具体化的计划。我打算从十一月上旬开始，陆续完成下面这个很有挑战的学习计划。大致要同一时期一直做其中两个任务，但是，哪个先做，哪个后做，我此刻暂时不能确定下来。希望有跟我有相同兴趣的朋友来与我联系。我可以加以合理的调整。

为什么做不好做不成

那些凄凄戚戚，一心只‘想要朋友’的人，从来结交不到朋友。——C.S.Lewis, ‘四种爱’

我对生活一直有一个困惑，就是我发现下面的事情十分的普遍。当人们集中他们所能集中的一切力量，排他的做一件事情的时候，往往结果是做不好，这跟他们的预期恰恰相反，跟他们的投入相比十分不对称。从我整个成长的过程中，我在身边便看到了很多很多这样的例子。

在上大学之前，我身边的几乎全部的同学都把那么多精力投入到准备高考中去，包括那些所谓的‘学习不好’的孩子。他们也每天花费大量的时间学习。甚至我还了解到一个更极端一些的例子，这个一个跟我差不多同龄的孩子。他因为高考失利而反复重考，有时候会读上几年，但是遇到不如意又走回头路。去年，他放弃一切重新参加高考，依然成绩不甚理想。而他的母亲一直采取陪读的策略。结果，数年的大好光阴全部浪费在准备一个无聊考试之上。这基本上是一个悲剧。我之前想这个问题的时候，很简单的把原因归结到学习效率低下和长期的逆反情绪。当然这些都是正确的原因，但是并不是很深层次的原因。