Found problems: 54
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 $z=q(2\cos(2\pi/n))$.
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 $$d!d((n-1)!+1) + n(d!-1) \equiv 0 \hspace{0.2cm} (\bmod n(n+d)).$$
2012 CIIM, Problem 5
Let $D=\{0,1,\dots,9\}$. A direction function for $D$ is a function $f:D \times D \to \{0,1\}.$
A real $r\in [0,1]$ is compatible with $f$ if it can be written in the form $$r = \sum_{j=1}^{\infty} \frac{d_j}{10^j}$$ with $d_j \in D$ and $f(d_j,d_{j+1})=1$ for every positive integer $j$.
Determine the least integer $k$ such that for any direction fuction $f$, if there are $k$ compatible reals with $f$ then there are infinite reals compatible with $f$.
2017 CIIM, Problem 3
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 $f(k) = f(k + 1).$
2018 CIIM, Problem 1
Show that there exists a $2 \times 2$ matrix of order 6 with rational entries, such that the sum of its entries is 2018.
Note: The order of a matrix (if it exists) is the smallest positive integer $n$ such that $A^n = I$, where $I$ is the identity matrix.
2013 CIIM, Problem 5
Let $A,B$ be $n\times n$ matrices with complex entries. Show that there exists a matrix $T$ and an invertible matrix $S$ such that \[ B=S(A+T)S^{-1}\ -T \iff \operatorname{tr}(A) = \operatorname{tr}(B) \]
2018 CIIM, Problem 4
Let $\alpha < 0 < \beta$ and consider the polynomial $f(x) = x(x-\alpha)(x-\beta)$. Let $S$ be the set of real numbers $s$ such that $f(x) - s$ has three different real roots. For $s\in S$, let $p(x)$ the product of the smallest and largest root of $f(x)-s$. Determine the smallest possible value that $p(s)$ for $s\in S$.
2010 CIIM, Problem 1
Given two vectors $v = (v_1,\dots,v_n)$ and $w = (w_1\dots,w_n)$ in $\mathbb{R}^n$, lets define $v*w$ as the matrix in which the element of row $i$ and column $j$ is $v_iw_j$. Supose that $v$ and $w$ are linearly independent. Find the rank of the matrix $v*w - w*v.$
2012 CIIM, Problem 2
A set $A\subset \mathbb{Z}$ is "padre" if whenever $x,y \in A$ with $x\leq y$ then also $2y -x \in A$. Prove that if $A$ is "padre", $0,a,b \in A$ with $0< a < b$ and $d = g.c.d(a,b)$ then \[a+b-3d, a+b-2d \in A.\]
2014 CIIM, Problem 5
A analityc function $f:\mathbb{C}\to\mathbb{C}$ is call interesting if $f(z)$ is real along the parabola $Re (z) = (Im (z))^2$.
a) Find an example of a non constant interesting function.
b) Show that every interesting function $f$ satisfy that $f'(-3/4) = 0.$
2012 CIIM, Problem 3
Let $a,b,c,$ the lengths of the sides of a triangle. Prove that \[\sqrt{\frac{(3a+b)(3b+a)}{(2a+c)(2b+c)}} + \sqrt{\frac{(3b+c)(3c+b)}{(2b+a)(2c+a)}} + \sqrt{\frac{(3c+a)(3a+c)}{(2c+b)(2a+b)}} \geq 4.\]
2014 Contests, Problem 2
Let $n$ be an integer and $p$ a prime greater than 2. Show that: $$(p-1)^nn!|(p^n-1)(p^n-p)(p^n-p^2)\cdots(p^n-p^{n-1}).$$
2014 CIIM, 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 $|\mathcal{A}| \leq 2^{n-2}.$
2012 CIIM, Problem 6
Let $n \geq 2$ and $p(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$ a polynomial with real coefficients. Show that if there exists a positive integer $k$ such that $(x-1)^{k+1}$ divides $p(x)$ then \[\sum_{j=0}^{n-1}|a_j| > 1 +\frac{2k^2}{n}.\]
2009 CIIM, Problem 4
Let $m$ be a line in the plane and $M$ a point not in $m$. Find the locus of the focus of the parabolas with vertex $M$ that are tangent to $m$.
2012 CIIM, Problem 1
For each positive integer $n$ let $A_n$ be the $n \times n$ matrix such that its $a_{ij}$
entry is equal to ${i+j-2 \choose j-1}$ for all $1\leq i,j \leq n.$ Find the determinant of $A_n$.
2015 CIIM, Problem 2
Find all polynomials $P(x)$ with real coefficients that satisfy the identity $$P(x^3-2)=P(x)^3-2,$$ for every real number $x$.
2011 CIIM, Problem 6
Let $\Gamma$ be the branch $x> 0$ of the hyperbola $x^2 - y^2 = 1.$ Let $P_0, P_1,..., P_n$ different points of $\Gamma$ with $P_0 = (1, 0)$ and $P_1 = (13/12, 5/12)$. Let $t_i$ be the tangent line to $\Gamma$ at $P_i$. Suppose that for all $i \geq 0$ the area of the region bounded by $t_i, t_{i +1}$ and $\Gamma$ is a constant independent of $i$. Find the coordinates of the points $P_i$.
2010 CIIM, Problem 2
In one side of a hall there are $2N$ rooms numbered from 1 to $2N$. In each room $i$ between 1 and $N$ there are $p_i$ beds. Is needed to move every one of this beds to the roms from $N+ 1$ to $2N$, in such a way that for every $j$ between $N+1$ and $2N$ the room $j$ will have $p_j$ beds. Supose that each bed can be move once and the price of moving a bed from room $i$ to room $j$ is $(i-j)^2$.
Find a way to move every bed such that the total cost is minimize.
Note: The numbers $p_i$ are given and satisfy that $p_1 + p_2 + \cdots + p_N = p_{N+1} + p_{N+2} + \cdots+ p_{2N}.$
2014 Contests, 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 $(c,d) \subset[2013,2014].$
2013 CIIM, Problem 3
Given a set of boys and girls, we call a pair $(A,B)$ amicable if $A$ and $B$ are friends. The friendship relation is symmetric. A set of people is affectionate if it satisfy the following conditions:
i) The set has the same number of boys and girls.
ii) For every four different people $A,B,C,D$ if the pairs $(A,B),(B,C),(C,D)$ and $(D,A)$ are all amicable, then at least one of the pairs $(A,C)$ and $(B,D)$ is also amicable.
iii) At least $\frac{1}{2013}$ of all boy-girl pairs are amicable.
Let $m$ be a positive integer. Prove that there exists an integer $N(m)$ such that if a affectionate set has al least $N(m)$ people, then there exists $m$ boys that are pairwise friends or $m$ girls that are pairwise friends.
2015 CIIM, Problem 6
Show that there exists a real $C > 1$ that satisfy the following property: if $n > 1$ and $a_0 < a_1 < \cdots < a_n$ are positive integers such that $\frac{1}{a_0},\frac{1}{a_1},\dots,\frac{1}{a_n}$ are in arithmetic progression, then $a_0 > C^n.$
2017 CIIM, Problem 1
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 $p(w) = 0.$
2018 CIIM, Problem 5
Consider the transformation $$T(x,y,z) = (\sin y + \sin z - \sin x,\sin z + \sin x - \sin y,\sin x +\sin y -\sin z).$$
Determine all the points $(x,y,z) \in [0,1]^3$ such that $T^n(x,y,z) \in [0,1]^3,$ for every $n \geq 1$.
2011 CIIM, Problem 3
Let $f(x)$ be a rational function with complex coefficients whose denominator does not have multiple roots. Let $u_0, u_1,... , u_n$ be the complex roots of $f$ and $w_1, w_2,..., w_m$ be the roots of $f'$. Suppose that $u_0$ is a simple root of $f$. Prove that
\[ \sum_{k=1}^m \frac{1}{w_k - u_0} = 2\sum_{k = 1}^n\frac{1}{u_k - u_0}.\]