ONMIPA-PT; responding to the last post: locating Indonesian mathematics student in the global mathematical society

Hello everybody. Well, responding to my last issue of Indonesian mathematical society (of which I’ve posted in Bahasa Indonesia), I would like to share my last 2 days experience. Yesterday and today, 30-31 April 2012; yes, the last two days on April; I was attending a regional mathematics competition, namely ONMIPA-PT (Olimpiade Nasional Matematika dan Ilmu Pengetahuan Alam Perguruan Tinggi); or in English: Collegiate Olympiad in Mathematics and Natural Sciences. Well, you should be familiar with high-school olympiad, such as IMO (mathematics), IBO (biology), IOI (computing), etc. Then ONMIPA-PT should be one of these stuffs, in which the participants are the undergraduate. A little information about ONMIPA-PT you should know. In Indonesia, we do not find many mathematics competition held for undergraduate. However, ONMIPA-PT should be the one of the most prestigious mathematics & natural science competition for undergraduate, since the program is held by Indonesian National Ministry of Education. As long as I know, among many other competitions in Indonesia, ONMIPA-PT has the best quality of the problems presented in the contest. That’s my short introduction to ONMIPA-PT. Well then, in this opportunity, I was attending the west java regional selection of ONMIPA-PT, in order to get to the national selection which will be held in May.

The main idea which I would like to discuss, is the material of the tests. The tests themselves consist of 5 section: real analysis, combinatorics, complex analysis, abstract algebra, and linear algebra. Although (as I’ve mentioned above) ONMIPA-PT has the best quality of the problems, I’ve found some problems there are equivalent to our regular mid term test problem in ITB. I don’t know, but I think that the difficulty of ONMIPA should be much much higher than regular mid term test problem. I would like to discuss to everyone about how is the quality of the problems? Are they too easy, or too difficult, or just so so? I really appreciate any opinion about my issues. Below I present you 1 problem of each section, in which I consider the most difficult among other problems.

Real Analysis

Given a real-valued function f is differentiable on [a,b]. Show that for each \gamma \in [f'(a),f'(b)] ;WLOG, f'(a)<f'(b), there is c \in (a,b) such that f'(c)=\gamma.

Combinatorics

Given any 7 different real numbers. Show that there are always 2 of them, namely a and b such that 0<\frac{a-b}{1+ab} < \sqrt{3}.

Complex Analysis

Given f is a complex-valued function. f and its conjugate are analytic in a connected domain. Show that f must be constant.

Abstract Algebra

Let G be any set having an associative multiplication operation. Suppose there is an element e \in G enjoying the following properties:

  1. For each g \in G, ge=g
  2. For each g \in G there always g' \in G such that gg'=e

Show that G is a group.

Linear Algebra

Let A \in \mathbb{R}^{2013 \times 2013} with A=[a_{ij}]. The value of a_{ij} is defined as follows: a_{ij}=1 if j>i or j=i-1a_{ij}=0 if j<i-1 or j=i. Calculate the determinant of A.

Well, those are the problems of last 2 day’s ONMIPA-PT. Any comment about the difficulty level of the problem is extremely welcome!

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