Let $ \sigma(\cdot): \mathbb{N}_0 \to \mathbb{N}_0$ be the function from every positive integer $ n$ to the sum of divisors $ \sum_{d \mid n}{d}$ (i.e. $ \sigma(6) = 6 + 3 + 2 + 1$ and $ \sigma(8) = 8 + 4 + 2 + 1$). Find all primes $ p$ such that $ p \mid \sigma(p - 1)$. (Salvatore Tringali)

Define $ \phi$ the positive real root of $ x^2 - x - 1$ and let $ a,b,c,d$ be positive real numbers such that $ (a + 2b)^2 = 4c^2 + 1$. Show that $ \displaystyle 2d^2 + a^2\left(\phi - \frac {1}{2}\right) + b^2\left(\frac {1}{\phi - 1} + 2\right) + 2 \ge 4(c - d) + 2\sqrt {d^2 + 2d}$ and find all cases of equality. (A.Naskov)

Let a cyclic quadrilateral $ ABCD$, $ AC \cap BD = E$ and let a circle $ \Gamma$ internally tangent to the arch $ BC$ (that not contain $ D$) in $ T$ and tangent to $ BE$ and $ CE$. Call $ R$ the point where the angle bisector of $ \angle ABC$ meet the angle bisector of $ \angle BCD$ and $ S$ the incenter of $ BCE$. Prove that $ R$, $ S$ and $ T$ are collinear. (Gabriel Giorgieri)

Let $ m$ a positive integer and $ p$ a prime number, both fixed. Define $ S$ the set of all $ m$-uple of positive integers $ \vec{v} = (v_1,v_2,\ldots,v_m)$ such that $ 1 \le v_i \le p$ for all $ 1 \le i \le m$. Define also the function $ f(\cdot): \mathbb{N}^m \to \mathbb{N}$, that associates every $ m$-upla of non negative integers $ (a_1,a_2,\ldots,a_m)$ to the integer $ \displaystyle f(a_1,a_2,\ldots,a_m) = \sum_{\vec{v} \in S} \left(\prod_{1 \le i \le m}{v_i^{a_i}} \right)$. Find all $ m$-uple of non negative integers $ (a_1,a_2,\ldots,a_m)$ such that $ p \mid f(a_1,a_2,\ldots,a_m)$. (Pierfrancesco Carlucci)

Let $ X: = \{x_1,x_2,\ldots,x_{29}\}$ be a set of $ 29$ boys: they play with each other in a tournament of Pro Evolution Soccer 2009, in respect of the following rules: i) every boy play one and only one time against each other boy (so we can assume that every match has the form $ (x_i \text{ Vs } x_j)$ for some $ i \neq j$); ii) if the match $ (x_i \text{ Vs } x_j)$, with $ i \neq j$, ends with the win of the boy $ x_i$, then $ x_i$ gains $ 1$ point, and $ x_j$ doesn’t gain any point; iii) if the match $ (x_i \text{ Vs } x_j)$, with $ i \neq j$, ends with the parity of the two boys, then $ \frac {1}{2}$ point is assigned to both boys. (We assume for simplicity that in the imaginary match $ (x_i \text{ Vs } x_i)$ the boy $ x_i$ doesn’t gain any point). Show that for some positive integer $ k \le 29$ there exist a set of boys $ \{x_{t_1},x_{t_2},\ldots,x_{t_k}\} \subseteq X$ such that, for all choice of the positive integer $ i \le 29$, the boy $ x_i$ gains always a integer number of points in the total of the matches $ \{(x_i \text{ Vs } x_{t_1}),(x_i \text{ Vs } x_{t_2}),\ldots, (x_i \text{ Vs } x_{t_k})\}$. (Paolo Leonetti)

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)

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)

Find all $ (x,y,z) \in \mathbb{Z}^3$ such that $ x^3 - 5x = 1728^{y}\cdot 1733^z - 17$. (Paolo Leonetti)

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)

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)

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 2 rounds. 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 17:00 of 28 september and it ends on 17:00 of 30 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@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! {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

final round 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