CIIM 2017

Determine all the complex numbers $w = a + bi$ with $a, b \in \mathbb{R}$ , such that there exists a polinomial $p(z)$ whose coefficients are real and positive such that $p(w) = 0.$

Let $f :\mathbb{R} \to \mathbb{R}$ a derivable function such that $f(0) = 0$ and $|f'(x)| \leq |f(x)\cdot log |f(x)||$ for every $x \in \mathbb{R}$ such that $0 < |f(x)| < 1/2.$ Prove that $f(x) = 0$ for every $x \in \mathbb{R}$ .

Let $G$ be a finite abelian group and $f :\mathbb{Z}^+ \to G$ a completely multiplicative function (i.e. $f(mn) = f(m)f(n)$ for any positive integers $m, n$ ). Prove that there are infinitely many positive integers $k$ such that $f(k) = f(k + 1).$

Let $m, n$ be positive integers and $a_1,\dots , a_m, b_1, \dots , b_n$ positive real numbers such that for every positive integer $k$ we have that $$(a_1^k + \cdots + a^k_m) - (b^k_1 + \cdots + b^k_n) \leq CkN,$$ for some fix $C$ and $N$ . Show that there exists $l \leq m, n$ and permutations $\sigma$ of $\{1, \dots , m\}$ and $\tau$ of $\{1,\dots , n\}$ , such that

1. $a\sigma(i) = b\tau(i)$ for $1 \leq i \leq l,$ 2. $a\sigma(i) , b\tau(i) \leq 1$ for $i > l.$

Let $\mathcal{S}$ be a set of integers. Given a real positive $r$, we say that $\mathcal{S}$ is a $r$ - discerning, if for any pair $m, n > 1$ of distinct integers such that $\left| \frac{m-n}{m+n} \right| < r$, there exists $a \in \mathcal{S}$ and $k \geq 1$ such that $a^k$ divides $m$ but not $n$, or $a^k$ divides $n$ but not $m$

1. Show that for every $r > 0$ every $r$ - discerning set contains an infinite number of primes.

2. For every $r > 0$ determine the maximal possible cardinality of $\mathcal{P} \backslash \mathcal{S}$ where $\mathcal{P}$ is the set of primes and $\mathcal{S} \subseteq \mathcal{P}$ is a $r$ - discerning set.

Let $G$ be a simple, connected and finite grafo. A hunter and an invisible rabbit play in the graph $G$ . The rabbit is initially in a vertex $w_0$ . In the $k$ -th turn (for $k \geq 0$ ) the hunter picks freely a vertex $v_k$ . If $v_k = w_k$ , the rabbit is capture and the game ends. If not, the rabbit moves invisibly by an edge of $w_k$ to $w_{k+1}$ ( $w_k$ and $w_{k+1}$ are adjacent and therefore distinct) and the game continues. The hunter knows these rules and the graph $G$ . After the $k-$ th turn he knows that $w_k \not = v_k$ , but he gets no more information. Characterize the graphs $G$ such that the hunter has an strategy that guaranties that he can capture the rabbit in at most $N$ turns for some positive integer $N$ . Here $N$ must depend only on $G$ and the strategy should work independently of the initial position and trajectory of the rabbit.