(a) Prove that in the set $ S=\{2008,2009,. . .,4200\}$ there are $ 5^3$ elements such that any three of them are not in arithmetic progression. (b) Bonus: Try to find a smaller integer $ n \in (2008,4200)$ such that in the set $ S'=\{2008,2009,...,n\}$ there are $ 5^3$ elements such that any three of them are not in arithmetic progression.

Let $ ABCD$ be a cyclic quadrilateral with $ AB>CD$ and $ BC>AD$. Take points $ X$ and $ Y$ on the sides $ AB$ and $ BC$, respectively, so that $ AX=CD$ and $ AD=CY$. Let $ M$ be the midpoint of $ XY$. Prove that $ AMC$ is a right angle.

Let $ a,b,c$ be three pairwise distinct real numbers such that $ a+b+c=6=ab+bc+ca-3$. Prove that $ 0<abc<4$.

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.

Let $ \{a_n\}_{n \in \mathbb{N}_0}$ be a sequence defined as follows: $ a_1=0$, $ 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$.

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'B'C'$. Prove that $ AA',BB'$ and $ CC'$ are concurrent.

Let $ a,b,c$ positive reals such that $ ab + bc + ca = 3$, show that: $ \displaystyle a^2 + b^2 + c^2 + 3 \ge \frac {a(3 + bc)^2}{(c + b)(b^2 + 3)} + \frac {b(3 + ca)^2}{(a + c)(c^2 + 3)} + \frac {c(3 + ab)^2}{(b + a)(a^2 + 3)}$ (Anass BenTaleb, Ali Ben Bari High School - Taza,Morocco)

Find all non-negative integers $ x,y,z$ such that $ 5^x + 7^y = 2^z$. (Daniel Kohen, University of Buenos Aires - Buenos Aires,Argentina)

Let $ 0 < a_1 < a_2 < a_3 < ... < a_{10000} < 20000$ be integers such that $ gcd(a_i,a_j) < a_i, \forall i < j$ ; is $ 500 < a_1$ (always) true ? (own)

Consider the sequence of integer such that: $ a_1 = 2$ $ a_2 = 5$ $ a_{n + 1} = (2 - n^2)a_n + (2 + n^2)a_{n - 1}, \forall n\ge 2$ Find all triplies $ (x,y,z) \in \mathbb{N}^3$ such that $ a_xa_y = a_z$.

Let $ a_1,a_2,...,a_n$ with arithmetic mean equals zero; what is the value of: $ \sum_{j=1}^n{\frac{1}{a_j(a_j+a_{j+1})(a_j+a_{j+1}+a_{j+2})...(a_j+a_{j+1}+...+a_{j+n-2})}}$ , where $ a_{n+k}=a_k$ ?

round 1 Hi guys, here's the topic of the "Oliforum contest", the competition for training that i'm arranging from some weeks . i'll try to sumarize all details about it, be carefully. 1- The contest is made of some rounds(I think 3 max). 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. 4- All begins from italian-oliforum and the classification is unique. 5- The first round starts on 12:00 of 20 september and it ends on 24:00 of 21 seprember, 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 21/21 points. 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}. Later, 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 Thanks for the attentions And good luck Paolo Leonetti Italian version

round 2 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

final round Still hi guys, here's the topic of the "Oliforum contest", the competition for training that started some months ago . You can find problem, solutions and ranks in the following links: first and second round. i'll sumarize all details about it, be carefully. 1- The contest is made of 3 rounds: this is the last one. 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. 4- All begins from italian-oliforum and the final rank is unique. You can find the partial rank at the end of second round at this link. 5- The round starts on 12:00 of 6 december and it ends on 24:00 of 21 seprember, so saturday and sunday (you have 36 hours in total!). 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 21/21 points. 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}. Later, 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 mp please, for clearity of all I'll put below this topic the list of problems on the morning of saturday 6 december, 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. Thanks for the attentions And good luck Paolo Leonetti Italian version