Found problems: 823
2004 USAMO, 5
Let $a, b, c > 0$. Prove that $(a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \geq (a + b + c)^3$.
1991 Vietnam Team Selection Test, 3
Let $\{x\}$ be a sequence of positive reals $x_1, x_2, \ldots, x_n$, defined by: $x_1 = 1, x_2 = 9, x_3=9, x_4=1$. And for $n \geq 1$ we have:
\[x_{n+4} = \sqrt[4]{x_{n} \cdot x_{n+1} \cdot x_{n+2} \cdot x_{n+3}}.\]
Show that this sequence has a finite limit. Determine this limit.
2005 IberoAmerican Olympiad For University Students, 2
Let $A,B,C$ be real square matrices of order $n$ such that $A^3=-I$, $BA^2+BA=C^6+C+I$ and $C$ is symmetric. Is it possible that $n=2005$?
2017 Romania National Olympiad, 3
Let be a natural number $ n\ge 2 $ and two $ n\times n $ complex matrices $ A,B $ that satisfy $ (AB)^3=O_n. $
Does this imply that $ (BA)^3=O_n ? $
2002 China Western Mathematical Olympiad, 4
Let $ n$ be a positive integer, let the sets $ A_{1},A_{2},\cdots,A_{n \plus{} 1}$ be non-empty subsets of the set $ \{1,2,\cdots,n\}.$ prove that there exist two disjoint non-empty subsets of the set $ \{1,2,\cdots,n \plus{} 1\}$: $ \{i_{1},i_{2},\cdots,i_{k}\}$ and $ \{j_{1},j_{2},\cdots,j_{m}\}$ such that $ A_{i_{1}}\cup A_{i_{2}}\cup\cdots\cup A_{i_{k}} \equal{} A_{j_{1}}\cup A_{j_{2}}\cup\cdots\cup A_{j_{m}}$.
2005 Moldova Team Selection Test, 3
For an ${n\times n}$ matrix $A$, let $X_{i}$ be the set of entries in row $i$, and $Y_{j}$ the set of entries in column $j$, ${1\leq i,j\leq n}$. We say that $A$ is [i]golden[/i] if ${X_{1},\dots ,X_{n},Y_{1},\dots ,Y_{n}}$ are distinct sets. Find the least integer $n$ such that there exists a ${2004\times 2004}$ golden matrix with entries in the set ${\{1,2,\dots ,n\}}$.
2006 Harvard-MIT Mathematics Tournament, 8
In how many ways can we enter numbers from the set $\{1,2,3,4\}$ into a $4\times 4$ array so that all of the following conditions hold?
(a) Each row contains all four numbers.
(b) Each column contains all four numbers.
(c) Each "quadrant" contains all four numbers. (The quadrants are the four corner $2\times 2$ squares.)
2016 Vietnam Team Selection Test, 6
Given $16$ distinct real numbers $\alpha_1,\alpha_2,...,\alpha_{16}$. For each polynomial $P$, denote \[ V(P)=P(\alpha_1)+P(\alpha_2)+...+P(\alpha_{16}). \] Prove that there is a monic polynomial $Q$, $\deg Q=8$ satisfying:
i) $V(QP)=0$ for all polynomial $P$ has $\deg P<8$.
ii) $Q$ has $8$ real roots (including multiplicity).
2011 District Olympiad, 2
Consider the matrices $A\in \mathcal{M}_{m,n}(\mathbb{C})$ and $B\in \mathcal{M}_{n,m}(\mathbb{C})$ with $n\le m$. It is given that $\text{rank}(AB)=n$ and $(AB)^2=AB$.
a)Prove that $(BA)^3=(BA)^2$.
b)Find $BA$.
1965 Miklós Schweitzer, 3
Let $ a,b_0,b_1,b_2,...,b_{n\minus{}1}$ be complex numbers, $ A$ a complex square matrix of order $ p$, and $ E$ the unit matrix of order $ p$. Assuming that the eigenvalues of $ A$ are given, determine the eigenvalues of the matrix
\[ B\equal{}\begin{pmatrix} b_0E&b_1A&b_2A^2&\cdots&b_{n\minus{}1}A^{n\minus{}1} \\
ab_{n\minus{}1}A^{n\minus{}1}&b_0E&b_1A&\cdots&b_{n\minus{}2}A^{n\minus{}2}\\
ab_{n\minus{}2}A^{n\minus{}2}&ab_{n\minus{}1}A^{n\minus{}1}&b_0E&\cdots&b_{n\minus{}3}A^{n\minus{}3}\\
\vdots&\vdots&\vdots&\ddots&\vdots&\\
ab_1A&ab_2A^2&ab_3A^3&\cdots&b_0E
\end{pmatrix}\quad\]
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 $A_n$.
1973 Miklós Schweitzer, 1
We say that the rank of a group $ G$ is at most $ r$ if every subgroup of $ G$ can be generated by at most $ r$ elements. Prove
that here exists an integer $ s$ such that for every finite group $ G$ of rank $ 2$ the commutator series of $ G$ has length less than $ s$.
[i]J. Erdos[/i]
2012 IMO Shortlist, A7
We say that a function $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is a metapolynomial if, for some positive integers $m$ and $n$, it can be represented in the form
\[f(x_1,\cdots , x_k )=\max_{i=1,\cdots , m} \min_{j=1,\cdots , n}P_{i,j}(x_1,\cdots , x_k),\]
where $P_{i,j}$ are multivariate polynomials. Prove that the product of two metapolynomials is also a metapolynomial.
2007 IberoAmerican Olympiad For University Students, 1
For each pair of integers $(i,k)$ such that $1\le i\le k$, the linear transformation $P_{i,k}:\mathbb{R}^k\to\mathbb{R}^k$ is defined as:
$P_{i,k}(a_1,\cdots,a_{i-1},a_i,a_{i+1},\cdots,a_k)=(a_1,\cdots,a_{i-1},0,a_{i+1},\cdots,a_k)$
Prove that for all $n\ge2$ and for every set of $n-1$ linearly independent vectors $v_1,\cdots,v_{n-1}$ in $\mathbb{R}^n$, there is an integer $k$ such that $1\le k\le n$ and such that the vectors $P_{k,n}(v_1),\cdots,P_{k,n}(v_{n-1})$ are linearly independent.
2023 Romania National Olympiad, 3
Let $n$ be a natural number $n \geq 2$ and matrices $A,B \in M_{n}(\mathbb{C}),$ with property $A^2 B = A.$
a) Prove that $(AB - BA)^2 = O_{n}.$
b) Show that for all natural number $k$, $k \leq \frac{n}{2}$ there exist matrices $A,B \in M_{n}(\mathbb{C})$ with property stated in the problem such that $rank(AB - BA) = k.$
2003 IMO Shortlist, 4
Let $x_1,\ldots, x_n$ and $y_1,\ldots, y_n$ be real numbers. Let $A = (a_{ij})_{1\leq i,j\leq n}$ be the matrix with entries \[a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases}\] Suppose that $B$ is an $n\times n$ matrix with entries $0$, $1$ such that the sum of the elements in each row and each column of $B$ is equal to the corresponding sum for the matrix $A$. Prove that $A=B$.
2013 SEEMOUS, Problem 2
Let $M,N\in M_2(\mathbb C)$ be two nonzero matrices such that
$$M^2=N^2=0_2\text{ and }MN+NM=I_2$$where $0_2$ is the $2\times2$ zero matrix and $I_2$ the $2\times2$ unit matrix. Prove that there is an invertible matrix $A\in M_2(\mathbb C)$ such that
$$M=A\begin{pmatrix}0&1\\0&0\end{pmatrix}A^{-1}\text{ and }N=A\begin{pmatrix}0&0\\1&0\end{pmatrix}A^{-1}.$$
2016 District Olympiad, 2
Show that:
$$ 2015\in\left\{ x_1+2x_2+3x_3\cdots +2015x_{2015}\big| x_1,x_2,\ldots ,x_{2015}\in \{ -2,3\}\right\}\not\ni 2016. $$
1996 Romania National Olympiad, 3
Let $A, B \in M_2(\mathbb{R})$ such as $det(AB+BA)\leq 0$. Prove that $$det(A^2+B^2)\geq 0$$
1950 Miklós Schweitzer, 6
Prove the following identity for determinants:
$ |c_{ik} \plus{} a_i \plus{} b_k \plus{} 1|_{i,k \equal{} 1,...,n} \plus{} |c_{ik}|_{i,k \equal{} 1,...,n} \equal{} |c_{ik} \plus{} a_i \plus{} b_k|_{i,k \equal{} 1,...,n} \plus{} |c_{ik} \plus{} 1|_{i,k \equal{} 1,...,n}$
2014 Lithuania Team Selection Test, 3
Given such positive real numbers $a, b$ and $c$, that the system of equations:
$ \{\begin{matrix}a^2x+b^2y+c^2z=1&&\\xy+yz+zx=1&&\end{matrix} $
has exactly one solution of real numbers $(x, y, z)$. Prove, that there is a triangle, which borders lengths are equal to $a, b$ and $c$.
2012 Gheorghe Vranceanu, 1
[b]a)[/b] Find all $ 2\times 2 $ complex matrices $ A $ which have the property that there are two complex numbers $ \alpha ,\gamma $ with $ \alpha \neq \text{tr} (A) $ or $ \gamma\neq \det (A) $ such that $ A^2-\alpha A+\gamma I=0. $
[b]b)[/b] Consider $ B\not\in\{ 0,I\} $ as a matrix having the property mentioned at [b]a).[/b]
Solve in the complex numbers the system $ xB-yI-B^2=xB^2-yI-B^4=0. $
[i]Adrian Troie[/i]
2000 Moldova National Olympiad, Problem 7
Prove that for any positive integer $n$ there exists a matrix of the form
$$A=\begin{pmatrix}1&a&b&c\\0&1&a&b\\0&0&1&a\\0&0&0&1\end{pmatrix},$$
(a) with nonzero entries,
(b) with positive entries,
such that the entries of $A^n$ are all perfect squares.
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 Junior Balkan Team Selection Tests - Romania, 4
The vertices of a regular $n$-gon are initially marked with one of the signs $+$ or $-$ . A [i]move[/i] consists in choosing three consecutive vertices and changing the signs from the vertices , from $+$ to $-$ and from $-$ to $+$.
[b]a)[/b] Prove that if $n=2015$ then for any initial configuration of signs , there exists a sequence of [i]moves[/i] such that we'll arrive at a configuration with only $+$ signs.
[b]b)[/b] Prove that if $n=2016$ , then there exists an initial configuration of signs such that no matter how we make the [i]moves[/i] we'll never arrive at a configuration with only $+$ signs.