Hi guys! here's the second round of the "Oliforum contest"! You can find rules and problems of first round here . i'll try to sum up again all details about it, please be carefully. 1- The contest is made of some rounds(I still think 3 in total). 2- Each round is made of only 3 problems, 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 (e.g. in the first round someone uses calculus or link at other solutions, please don't do that!) 4- All begins from italian-oliforum and the classification is unique (it will be made only of contestant with score $ > 7$ points ). 5- The second round starts on 12:00 of 4 october and it ends on 24:00 of 5 october, so saturday and sunday. 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- I’ll take in consideration time of arriving solutions only for the first place if there are more full score 21/21 points (as i did in first round). 8- Way of sending solutions. - You need to send solutions to both email address: pao_leo88@hotmail.it and leonettipaolo@gmail.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! I’ll estimate it! {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}. I'll put below this topic the list of problems on the morning of saturday 4 october, at 12:00 (+2GTM). Later, I'll make a list still 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 mp please, for clearity of this topic Thanks for the attentions And good luck Paolo Leonetti Italian version
Problem
Source:
Tags: calculus, email, LaTeX
04.10.2008 12:56
OLIFORUM CONTEST- ROUND 2 Problem 1 Let $ p>3$ be a prime. If $ p$ divides $ x$, prove that the equation $ x^2-1=y^p$ does not have positive integer solutions. Problem 2 Let $ \{a_n\}_{n \in \mathbb{N}_0}$ be a sequence defined as follows: $ a_1=0$, $ \displaystyle a_n=a_{[\frac{n}{2}]}+(-1)^{n(n+1)/2}$, where $ [x]$ denotes the floor function. For every $ k \ge 0$, find the number $ n(k)$ of positive integers $ n$ such that $ 2^k \le n < 2^{k+1}$ and $ a_n=0$. Problem 3 Let $ C_1,C_2$ and $ C_3$ be three pairwise disjoint circles. For each pair of disjoint circles, we define their internal tangent lines as the two common tangents which intersect in a point between the two centres. For each $ i,j$, we define $ (r_{ij},s_{ij})$ as the two internal tangent lines of $ (C_i,C_j)$. Let $ r_{12},r_{23},r_{13},s_{12},s_{13},s_{23}$ be the sides of $ ABCA^1B^1C^1$. Prove that $ AA^1,BB^1$ and $ CC^1$ are concurrent. (You can see the figure at this link) For any doubts contact me in mp e/o email Good work
Attachments:
olicontest2(2).pdf (47kb)
05.10.2008 02:38
Here the list of contestant who sent me solutions begins: Athinaios Broniran pak-man Akashil dima_ukraine kaymar 1991 mod_2 giove julio14 Rigel elendil sumita [..]
06.10.2008 02:45
You can also post your solutions here, waiting the gradation.. Problem 1 Problem 2 Problem 3
07.10.2008 23:49
Here it is the gradation (with scores $ \ge 6$ points, so at least one problem correct): I . Giove (10pp) II. sumita (9pp) III. ex-aequo: Akashnil , mod_2 (7pp) V. julio14 (6pp) Thanks for partecipating also to: Rigel pak-man Kaymar1991 broniran dima_ukraine elendil athinaios * Honorable mention to sumita: he's the only who solved the first problem * Honorable mention to giove : he's the only who gets at least one point on third problem (and getting nice observations about the first ptoblem) (in those days we'll send email with personal score to all contestants) Sum of partial scores (Iround+IIround with scores > 13 points) I: giove (31) II: sumita (29) III: Akashnil (28 ) IV: julio14 (26) V: Kaymar1991 (23) VI, ex-aequo: silouan, tomci0 (21) VIII: Rigel (18 ) IX: A Cave (16) X: mod_2 (14)
13.10.2008 18:01
Hi. Sorry for not participating to the 2nd round... I had no internet for 9 days because i moved to another city (where the University is) Does anyone know when the last round is?
15.10.2008 01:35
hi Nicknafplio! dont worry.. however next round will be on next month, now i'm too busy with exams, i'll contact for more details..