Hi guys, I'm Paolo and this is the second round of the second edition of Oliforum contest. You can see first round here. I copy below all rules, as always 1- The contest is made of 2 rounds, this is the last one. 2- Each round is made of some problems (from 3 to 5..), and i will give points 0 to 7: 6 for all correct solutions, 7 for only elegants and correct solutions (similar to ML contest). 3- It is tought for high-school student, and also for undergraduate or higher student, but you cannot use non-elementary-theory, otherwise the solution is invalid. 4- All begins from italian-oliforum and the classification is unique. 5- The first round starts on 15:00 of 16 september and it ends on 15:00 of 18 seprember, so you have 48 hours . NB. All times (date and hours) are according to ROME MERIDIAN (+2GTM), please attention! 6- Enrollment is not necessary, it sufficies to send me solution. 7- Time of arriving solution is not important. 8- Way of sending solutions. - You need to send solutions to both email address: pao_leo88 ML hotmail ML it and leonettipaolo ML gmail ML com in a unique pdf file (so, with attachment) of a right size; - The pdf file must be written with Latex (or a way that I can understand); - You rename the pdf file as your nickname (the same of yours in ML); - You need to write nothing in the email; - Try to be clear in solutions! {If (and only if) you are not able to create a pdf in latex model, you can send me a email in .doc format with solutions, as latex or in a way that i (and all correctors) can understand, or also with private message to me}. Later, together with texts, I'll make a list below this topic where i'll write all names of who sent me solutions in right way. For all doubts and questions, ask me in private messages Thanks for the attentions and good luck Paolo Leonetti
Problem
Source:
Tags: email, LaTeX, function, inequalities
16.10.2009 15:54
Problem 1. Find all non empty subset $ S$ of $ \mathbb{N}: = \{0,1,2,\ldots\}$ such that $ 0 \in S$ and exist two function $ h(\cdot): S \times S \to S$ and $ k(\cdot): S \to S$ which respect the following rules: i) $ k(x) = h(0,x)$ for all $ x \in S$ ii) $ k(0) = 0$ iii) $ h(k(x_1),x_2) = x_1$ for all $ x_1,x_2 \in S$. (Pierfrancesco Carlucci) Problem 2. Let a convex quadrilateral $ ABCD$ fixed such that $ AB = BC$, $ \angle ABC = 80, \angle CDA = 50$. Define $ E$ the midpoint of $ AC$; show that $ \angle CDE = \angle BDA$ (Paolo Leonetti) Problema 3. Find all $ (x,y,z) \in \mathbb{Z}^3$ such that $ x^3 - 5x = 1728^{y}\cdot 1733^z - 17$. (Paolo Leonetti) Problem 4. Let $ a,b,c$ be positive reals; show that $ \displaystyle a + b + c \leq \frac {bc}{b + c} + \frac {ca}{c + a} + \frac {ab}{a + b} + \frac {1}{2}\left(\frac {bc}{a} + \frac {ca}{b} + \frac {ab}{c}\right)$ (Darij Grinberg) Problem 5. Define the function $ g(\cdot): \mathbb{Z} \to \{0,1\}$ such that $ g(n) = 0$ if $ n < 0$, and $ g(n) = 1$ otherwise. Define the function $ f(\cdot): \mathbb{Z} \to \mathbb{Z}$ such that $ f(n) = n - 1024g(n - 1024)$ for all $ n \in \mathbb{Z}$. Define also the sequence of integers $ \{a_i\}_{i \in \mathbb{N}}$ such that $ a_0 = 1$ e $ a_{n + 1} = 2f(a_n) + \ell$, where $ \ell = 0$ if $ \displaystyle \prod_{i = 0}^n{\left(2f(a_n) + 1 - a_i\right)} = 0$, and $ \ell = 1$ otherwise. How many distinct elements are in the set $ S: = \{a_0,a_1,\ldots,a_{2009}\}$? (Paolo Leonetti) (For all doubts contact me in mp )
18.10.2009 21:48
Users who sent me solutions (which will be corrected as soon as possible): Ahwingsecretangent NickNafplio Giuseppe R mod_2 exodd Zhero azjps Bugi Palina.41 TBPL mavropnevma Ruby5099991 Maioc92 EUCLA dario2994 shoki kn skytin elendil Edit: Inequalities Master, Edit: grfield
19.10.2009 01:55
Problem1 Problem2 Problem3 Problem4 Problem5
19.10.2009 16:30
bboypa wrote: Users who sent me solutions (which will be corrected as soon as possible): Ahwingsecretangent NickNafplio Giuseppe R mod_2 exodd Zhero azjps Bugi Palina.41 TBPL mavropnevma Ruby5099991 Maioc92 EUCLA dario2994 shoki kn elendil I am pretty sure that I sent solutions, too..
19.10.2009 16:55
Oh yes, also you, very sorry!You was the first contestant who sent me solutions, and there were other mails between your..
19.10.2009 21:40
Here are the solutions
Attachments:
Oliforum contest 2009, 2 round.pdf (71kb)
30.10.2009 12:05
Hi everyone! Mr Paolo when are you going to post the final results?
30.10.2009 17:08
Hi! I'm very sorry for the delay, I have some exams to do, so I had no enough time to correct it.. However I hope in the next week to finish it
08.11.2009 16:21
when is round 2 starting?
10.11.2009 02:01
aadil, I'm sorry this is the second round, and it is ended.. Here there are ranks of second round: NickNafplio 7.7.7.6.0 shoki 7.6.7.7.0 TBPL 7.7.7.6.0 mavropnevma 7.7.4.7.1 grfield 7.7.4.6.0 dario2994 7.7.3.6.0 kn 7.7.2.6.0 Zhero 7.6.3.6.0 mod_2 7.0.7.7.0 Maioc92 7.6.2.6.0 azjps 7.7.7.0.0 exodd 7.6.2.6.0 Bugi 7.6.2.0.0 Palina.41 7.7.0.0.0 Inequalities Master 0.7.0.7.0 elendil 7.0.2.0.0 skytin 0.7.0.0.0 Ahwingsecretagent 7.0.0.0.0 GiuseppeR 2.0.2.3.0 Ruby5099991 0.0.0.0.2 EUCLA 0.0.0.0.1 And final ranks mavropnevma 28+26______54 TBPL 24+27______________51 NickNafplio 16+27________ 43 azjps 19+21_____________40 exodd 19+21_____________40 Zhero 17+22_____________39 dario2994 13+23_________36 kn 14+22_______________36 Maioc92 14+21___________35 mod_2 13+21____________34 shoki 0+27______________27 Bugi 12+15______________27 grfield 0+24______________24 Palina.41 7+14 ___________21 Inequalities Master 0+14 ___14 Ahwingsecretagent 7+7 ____14 elendil 0+9_______________9 GiuseppeR 1+7____________8 skytin 0+7________________7 geda 7+0_________________7 String 7+0________________7 Ruby5099991 0+2 _________2 EUCLA 0+1_______________1 Thanks to all to have partecipated, and I hope that we'll see again next year with third edition! Best regards to all Paolo Leonetti