MathDB

Oliforum Contest II 2009

Part of Oliforum Contest

Subcontests

(5)
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2

p divides f(a_1,a_2,...,a_m)

Let m m a positive integer and p p a prime number, both fixed. Define S S the set of all m m-uple of positive integers \vec{v} \equal{} (v_1,v_2,\ldots,v_m) such that 1≤vi≤p 1 \le v_i \le p for all 1≤i≤m 1 \le i \le m. Define also the function f(⋅):Nm→N f(\cdot): \mathbb{N}^m \to \mathbb{N}, that associates every m m-upla of non negative integers (a1,a2,…,am) (a_1,a_2,\ldots,a_m) to the integer \displaystyle f(a_1,a_2,\ldots,a_m) \equal{} \sum_{\vec{v} \in S} \left(\prod_{1 \le i \le m}{v_i^{a_i}} \right). Find all m m-uple of non negative integers (a1,a2,…,am) (a_1,a_2,\ldots,a_m) such that p∣f(a1,a2,…,am) p \mid f(a_1,a_2,\ldots,a_m). (Pierfrancesco Carlucci)

Omogeneus inequality by Daij Grinberg

Let a,b,c a,b,c be positive reals; show that \displaystyle a \plus{} b \plus{} c \leq \frac {bc}{b \plus{} c} \plus{} \frac {ca}{c \plus{} a} \plus{} \frac {ab}{a \plus{} b} \plus{} \frac {1}{2}\left(\frac {bc}{a} \plus{} \frac {ca}{b} \plus{} \frac {ab}{c}\right) (Darij Grinberg)
3
2
2
2
1
2

p divides sigma(p-1)

Let σ(⋅):N0→N0 \sigma(\cdot): \mathbb{N}_0 \to \mathbb{N}_0 be the function from every positive integer n n to the sum of divisors ∑d∣nd \sum_{d \mid n}{d} (i.e. \sigma(6) \equal{} 6 \plus{} 3 \plus{} 2 \plus{} 1 and \sigma(8) \equal{} 8 \plus{} 4 \plus{} 2 \plus{} 1). Find all primes p p such that p \mid \sigma(p \minus{} 1). (Salvatore Tringali)

All possible subset S of N

Find all non empty subset S S of \mathbb{N}: \equal{} \{0,1,2,\ldots\} such that 0∈S 0 \in S and exist two function h(⋅):S×S→S h(\cdot): S \times S \to S and k(⋅):S→S k(\cdot): S \to S which respect the following rules: i) k(x) \equal{} h(0,x) for all x∈S x \in S ii) k(0) \equal{} 0 iii) h(k(x_1),x_2) \equal{} x_1 for all x1,x2∈S x_1,x_2 \in S. (Pierfrancesco Carlucci)
5
2

I can obtain always integer number of points

Let X: \equal{} \{x_1,x_2,\ldots,x_{29}\} be a set of 29 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 (xi Vs xj) (x_i \text{ Vs } x_j) for some i≠j i \neq j); ii) if the match (xi Vs xj) (x_i \text{ Vs } x_j), with i≠j i \neq j, ends with the win of the boy xi x_i, then xi x_i gains 1 1 point, and xj x_j doesn’t gain any point; iii) if the match (xi Vs xj) (x_i \text{ Vs } x_j), with i≠j i \neq j, ends with the parity of the two boys, then 12 \frac {1}{2} point is assigned to both boys. (We assume for simplicity that in the imaginary match (xi Vs xi) (x_i \text{ Vs } x_i) the boy xi x_i doesn’t gain any point). Show that for some positive integer k≤29 k \le 29 there exist a set of boys {xt1,xt2,…,xtk}⊆X \{x_{t_1},x_{t_2},\ldots,x_{t_k}\} \subseteq X such that, for all choice of the positive integer i≤29 i \le 29, the boy xi x_i gains always a integer number of points in the total of the matches {(xi Vs xt1),(xi Vs xt2),…,(xi Vs xtk)} \{(x_i \text{ Vs } x_{t_1}),(x_i \text{ Vs } x_{t_2}),\ldots, (x_i \text{ Vs } x_{t_k})\}. (Paolo Leonetti)

Number of distinct element in {a_0,...,a_2009}

Define the function g(⋅):Z→{0,1} g(\cdot): \mathbb{Z} \to \{0,1\} such that g(n) \equal{} 0 if n<0 n < 0, and g(n) \equal{} 1 otherwise. Define the function f(⋅):Z→Z f(\cdot): \mathbb{Z} \to \mathbb{Z} such that f(n) \equal{} n \minus{} 1024g(n \minus{} 1024) for all n∈Z n \in \mathbb{Z}. Define also the sequence of integers {ai}i∈N \{a_i\}_{i \in \mathbb{N}} such that a_0 \equal{} 1 e a_{n \plus{} 1} \equal{} 2f(a_n) \plus{} \ell, where \ell \equal{} 0 if \displaystyle \prod_{i \equal{} 0}^n{\left(2f(a_n) \plus{} 1 \minus{} a_i\right)} \equal{} 0, and \ell \equal{} 1 otherwise. How many distinct elements are in the set S: \equal{} \{a_0,a_1,\ldots,a_{2009}\}? (Paolo Leonetti)