Found problems: 44
2013 CIIM, Problem 2
Consider a polinomial $p \in \mathbb{R}[x]$ of degree $n$ and with no real roots. Prove that $$\int_{-\infty}^{\infty}\frac{(p'(x))^2}{(p(x))^2+(p'(x))^2}dx$$ converges, and is less or equal than $n^{3/2}\pi.$
2014 CIIM, Problem 6
a) Let $\{x_n\}$ be a sequence with $x_n \in [0,1]$ for any $n$. Prove that there exists $C > 0$ such that for every positive integer $r$, there exists $m \geq 1$ and $n > m + r$ that satisfy $(n-m)|x_n-x_m| \leq C$.
b) Prove that for every $C > 0$, there exists a sequence $\{x_n\}$ with $x_n \in [0,1]$ for all $n$ and an integer $r$ such that, if $m \geq 1$ and $n > m+r$, then $(n-m)|x_n-x_m| > C.$
2019 CIIM, Problem 1
Determine all triples of integers $(x, y, z)$ that satisfy the equation
\[x^z + y^z = z.\]
2019 CIIM, Problem 3
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
\[\lim_{n\to\infty}\frac{|X_n|}{2^n} = 1.\]
2019 CIIM, Problem 2
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 $\{0,1\}^n$.
2010 CIIM, Problem 4
Let $f:[0,1] \to [0,1]$ a increasing continuous function, diferentiable in $(0,1)$ and with derivative smaller than 1 in every point. The sequence of sets $A_1,A_2,A_3,\dots$ is define as: $A_1 = f([0,1])$, and for $n \geq 2, A_n = f(A_{n-1}).$ Prove that $\displaystyle \lim_{n\to+\infty} d(A_n) = 0$, where $d(A)$ is the diameter of the set $A$.
Note: The diameter of a set $X$ is define as $d(X) = \sup_{x,y\in X} |x-y|.$
2012 CIIM, Problem 4
Let $f(x) = \frac{\sin(x)}{x}$ Find $$ \lim_{T\to\infty}\frac{1}{T}\int_0^T\sqrt{1+f'(x)^2}dx.$$
2016 CIIM, Problem 2
A boa of size $k$ is a graph with $k+1$ vertices $\{0,1,\dots,k-1,k\}$ and edges only between the vertices $i$ and $i+1$ for $0\leq i < k.$ The boa is place in a graph $G$ through a injection of graphs. (This is an injective function form the vertices of the boa to the vertices of the graph in such a way that if there is an edge between the vertices $x$ and $y$ in the boa then there must be an edge between $f(x)$ and $f(y)$ in $G$).
The Boa can move in the graph $G$ using to type of movement each time. If the boa is initially on the vertices $f(0),f(1),\dots,f(k)$ then it moves in one of the following ways:
(i) It choose $v$ a neighbor of $f(k)$ such that $v\not\in\{f(0),f(1),\dots,f(k-1)\}$ and the boa now moves to $f(0),f(1),\dots,f(k)$ with $f'(k)=v$ and $f'(i) = f(i+1)$ for $0 \leq i < k,$ or
(ii) It choose $v$ a neighbor of $f(0)$ such that $v\not\in\{f(1),f(2),\dots,f(k)\}$ and the boa now moves to $f(0),f(1),\dots,f(k)$ with $f'(0)=v$ and $f'(i) = f'(i-1)$ for $0 < i \leq k.$
Prove that if $G$ is a connected graph with diameter $d$, then it is possible to put a size $\lceil d/2 \rceil$ boa in $G$ such that the boa can reach any vertex of $G$.
2014 CIIM, Problem 4
Let $\{a_i\}$ be a strictly increasing sequence of positive integers. Define the sequence $\{s_k\}$ as $$s_k = \sum_{i=1}^{k}\frac{1}{[a_i,a_{i+1}]},$$ where $[a_i,a_{i+1}]$ is the least commun multiple of $a_i$ and $a_{i+1}$.
Show that the sequence $\{s_k\}$ is convergent.
2009 CIIM, Problem 3
Let $r > n$ be positive integers. A "good word" is an $n$-tuple $\langle a_1,\dots, a_n \rangle$ of distinct positive integers between 1 and $r$. A "play" consist of changing a integer $a_i$ of a good word, in such a way that the resulting word is still a good word. The distance between two good words $A= \langle a_1,\dots, a_n \rangle$ and $B = \langle b_1,\dots, b_n \rangle$ is the minimun number of plays needed to obtain B from A. Find the maximun posible distance between two good words.
2014 CIIM, Problem 1
Let $g:[2013,2014]\to\mathbb{R}$ a function that satisfy the following two conditions:
i) $g(2013)=g(2014) = 0,$
ii) for any $a,b \in [2013,2014]$ it hold that $g\left(\frac{a+b}{2}\right) \leq g(a) + g(b).$
Prove that $g$ has zeros in any open subinterval $(c,d) \subset[2013,2014].$
2014 Contests, Problem 3
Given $n\geq2$, let $\mathcal{A}$ be a family of subsets of the set $\{1,2,\dots,n\}$ such that, for any $A_1,A_2,A_3,A_4 \in \mathcal{A}$, it holds that $|A_1 \cup A_2 \cup A_3 \cup A_4| \leq n -2$.
Prove that $|\mathcal{A}| \leq 2^{n-2}.$
2010 CIIM, Problem 3
A set $X\subset \mathbb{R}$ has dimension zero if, for any $\epsilon > 0$ there exists a positive integer $k$ and intervals $I_1,I_2,...,I_k$ such that $X \subset I_1 \cup I_2 \cup \cdots \cup I_k$ with $\sum_{j=1}^k |I_j|^{\epsilon} < \epsilon$.
Prove that there exist sets $X,Y \subset [0,1]$ both of dimension zero, such that $X+Y = [0,2].$
2013 CIIM, Problem 4
Let $a_1,b_1,c_1,a_2,b_2,c_2$ be positive real number and $F,G:(0,\infty)\to(0,\infty)$ be to differentiable and positive functions that satisfy the identities: $$\frac{x}{F} = 1 + a_1x+ b_1y + c_1G$$ $$\frac{y}{G} = 1 + a_2x+ b_2y + c_2F.$$
Prove that if $0 < x_1 \leq x_2$ and $0 < y_2 \leq y_1$, then $F(x_1,x_2) \leq F(x_2,y_2)$ and $G(x_1,y_1) \geq G(x_2,y_2).$
2014 Contests, Problem 4
Let $\{a_i\}$ be a strictly increasing sequence of positive integers. Define the sequence $\{s_k\}$ as $$s_k = \sum_{i=1}^{k}\frac{1}{[a_i,a_{i+1}]},$$ where $[a_i,a_{i+1}]$ is the least commun multiple of $a_i$ and $a_{i+1}$.
Show that the sequence $\{s_k\}$ is convergent.
2019 CIIM, Problem 5
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).
2015 CIIM, Problem 1
Find the real number $a$ such that the integral $$\int_a^{a+8}e^{-x}e^{-x^2}dx$$ attain its maximum.
2013 CIIM, Problem 6
Let $(X,d)$ be a metric space with $d:X\times X \to \mathbb{R}_{\geq 0}$. Suppose that $X$ is connected and compact. Prove that there exists an $\alpha \in \mathbb{R}_{\geq 0}$ with the following property: for any integer $n > 0$ and any $x_1,\dots,x_n \in X$, there exists $x\in X$ such that the average of the distances from $x_1,\dots,x_n$ to $x$ is $\alpha$ i.e. $$\frac{d(x,x_1)+d(x,x_2)+\cdots+d(x,x_n)}{n} = \alpha.$$
2013 CIIM, Problem 1
Given two natural numbers $m$ and $n$, denote by $\overline{m.n}$ the number obtained by writing $m$ followed by $n$ after the decimal dot.
a) Prove that there are infinitely many natural numbers $k$ such that for any of
them the equation $\overline{m.n} \times \overline{n.m} = k$ has no solution.
b) Prove that there are infinitely many natural numbers $k$ such that for any of
them the equation $\overline{m.n} \times \overline{n.m} = k$ has a solution.
2011 CIIM, Problem 5
Let $n$ be a positive integer with $d$ digits, all different from zero. For $k = 0,. . . , d - 1$, we define $n_k$ as the number obtained by moving the last $k$ digits of $n$ to the beginning. For example, if $n = 2184$ then $n_0 = 2184, n_1 = 4218, n_2 = 8421, n_3 = 1842$. For $m$ a positive integer, define $s_m(n)$ as the number of values $k$ such that $n_k$ is a multiple of $m.$ Finally, define $a_d$ as the number of integers $n$ with $d$ digits all nonzero, for which $s_2 (n) + s_3 (n) + s_5 (n) = 2d.$
Find \[\lim_{d \to \infty} \frac{a_d}{5^d}.\]
2010 CIIM, Problem 6
A group is call locally cyclic if any finitely generated subgroup is cyclic. Prove that a locally cyclic group is isomorphic to one of its proper subgroups if and only if it's isomorphic to a proper subgroup of the rational numbers with the adition.
2019 CIIM, Problem 6
Determine all the injective functions $f : \mathbb{Z}_+ \to \mathbb{Z}_+$, such that for each pair of integers $(m, n)$ the following conditions hold:
$a)$ $f(mn) = f(m)f(n)$
$b)$ $f(m^2 + n^2) \mid f(m^2) + f(n^2).$
2009 CIIM, Problem 6
Let $\epsilon$ be an $n$-th root of the unity and suppose $z=p(\epsilon)$ is a real number where $p$ is some polinomial with integer coefficients. Prove there exists a polinomial $q$ with integer coefficients such that $z=q(2\cos(2\pi/n))$.
2009 CIIM, Problem 5
Let $f:\mathbb{R} \to \mathbb{R}$, such that
i) For all $a \in \mathbb{R}$ and all $\epsilon > 0$, exists $\delta > 0$ such that $|x-a| < \delta \Rightarrow f(x) < f(a) + \epsilon.$
ii) For all $b\in \mathbb{R}$ and all $\epsilon > 0$, exists $x,y \in \mathbb{R}$ with $ b - \epsilon < x < b < y < b + \epsilon$, such that $|f(x)-f(b)|< \epsilon$ and $|f(y)-f(b)| < \epsilon.$
Prove that if $f(a) < d < f(d)$ there exists $c$ with $a < c < b$ or $b < c < a$ such that $f(c) = d$.
2010 CIIM, Problem 5
Let $n,d$ be integers with $n,k > 1$ such that $g.c.d(n,d!) = 1$. Prove that $n$ and $n+d$ are primes if and only if $$d!d((n-1)!+1) + n(d!-1) \equiv 0 \hspace{0.2cm} (\bmod n(n+d)).$$