CIIM 2019

Determine all triples of integers $(x, y, z)$ that satisfy the equation $$x^z + y^z = z.$$

Consider the set $$\{0, 1\}^n = \{X = (x_1, x_2,\dots , x_n) : x_i \in \{0, 1\}, 1 \leq i \leq n\}.$$ We say that $X > Y$ if $X \neq Y$ and the following $n$ inequalities are satisfy $$x_1 \geq y_1, x_1 + x_2 \geq y_1 + y_2,\dots , x_1 + x_2 + \cdots + x_n \geq y_1 + y_2 + \cdots + y_n.$$ We define a chain of length $k$ as a subset ${Z_1,\dots , Z_k} \subseteq \{0, 1\}^n$ of distinct elements such that $Z_1 > Z_2 > \cdots > Z_k.$ Determine the lenght of longest chain in $\{0,1\}^n$ .

Let $\{a_n\}_{n\in \mathbb{N}}$ a sequence of non zero real numbers. For $m \geq 1$ , we define: $$X_m = \left\{X \subseteq \{0, 1,\dots, m - 1\}: \left|\sum_{x\in X} a_x \right| > \dfrac{1}{m}\right\}.$$ Show that $$\lim_{n\to\infty}\frac{|X_n|}{2^n} = 1.$$

Let $(G, *)$ a group of $n > 1$ elements, and let $g \in G$ be an element distinct from the identity. Ana and Bob play with the group $G$ on the following way: Starting with Ana and playing alternately, each player selects an element of $G$ that has not been selected before, until each element of $G$ have been selected or a player have selected the elements $a$ and $a * g$ for some $a \in G$ . In that case it is said that the player loses and his opponent wins. $a)$ If $n$ is odd, show that, independent of element $g$ , one of the two players has a winning strategy and determines which player possesses such a strategy. $b)$ If $n$ is even, show that there exists an element $g \in G$ for which none of the players has a winning strategy.

Note: A group $(G, *)$ es a set $G$ together with a binary operation $* : G\times G \to G$ that satisfy the following properties $(i)$ $*$ is asociative: $\forall a, b, c \in G (a * b) * c = a * (b * c)$ ; $(ii)$ there exists an identity element $e \in G$ such that $\forall a \in G, a *e = e * a = a;$ $(iii)$ there exists inverse elements: $\forall a \in G \exists a^{-1} \in G$ such that $a*a^{-1} = a^{-1} *a = e.$

Let $\{k_1, k_2, \dots , k_m\}$ a set of $m$ integers. Show that there exists a matrix $m \times m$ with integers entries $A$ such that each of the matrices $A + k_jI, 1 \leq j \leq m$ are invertible and their entries have integer entries (here $I$ denotes the identity matrix).

Determine all the injective functions $f : \mathbb{Z}_+ \to \mathbb{Z}_+$, such that for each pair of integers $(m, n)$ the following conditions hold:

$a)$ $f(mn) = f(m)f(n)$ $b)$ $f(m^2 + n^2) \mid f(m^2) + f(n^2).$