This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 638

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]

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.

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 $A=B$.

1988 French Mathematical Olympiad, Problem 1

Let us consider a matrix $T$ with n rows denoted $1,\ldots,n$ and $p$ columns $1,\ldots,p$. Its entries $a_{ik}~(1\le i\le n,1\le k\le p)$ are integers such that $1\le a_{ik}\le N$, where $N$ is a given natural number. Let $E_i$ be the set of numbers that appear on the $i$-th row. Answer question (a) or (b). (a) Assume $T$ satisfies the following conditions: $(1)$ $E_i$ has exactly $p$ elements for each $i$, and $(2)$ all $E_i$'s are mutually distinct. Let $m$ be the smallest value of $N$ that permits a construction of such an $n\times p$ table $T$. i. Compute $m$ if $n=p+1$. ii. Compute $m$ if $n=10^{30}$ and $p=1998$. iii. Determine $\lim_{n\to\infty}\frac{m^p}n$, where $p$ is fixed. (b) Assume $T$ satisfies the following conditions instead: $(1)$ $p=n$, $(2)$ whenever $i,k$ are integers with $i+k\le n$, the number $a_{ik}$ is not in the set $E_{i+k}$. i. Prove that all $E_i$'s are mutually distinct. ii. Prove that if $n\ge2^q$ for some integer $q>0$, then $N\ge q+1$. iii. Let $n=2^r-1$ for some integer $r>0$. Prove that $N\ge r$ and show that there is such a table with $N=r$.

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 $$M=A\begin{pmatrix}0&1\\0&0\end{pmatrix}A^{-1}\text{ and }N=A\begin{pmatrix}0&0\\1&0\end{pmatrix}A^{-1}.$$

1985 Traian Lălescu, 1.4

Without calculating the value of the determinant $$ \begin{vmatrix}1 &1 &3& 1\\1& 2& 3 &5\\ 3& 0& 5& 5\\ 0& a& -11a& a^{13}+9a\end{vmatrix} , $$ show that it is divisible by $ 26, $ for any integer $ a. $

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]

1999 IMO, 3

Let $n$ be an even positive integer. We say that two different cells of a $n \times n$ board are [b]neighboring[/b] if they have a common side. Find the minimal number of cells on the $n \times n$ board that must be marked so that any cell (marked or not marked) has a marked neighboring cell.

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).

2024 Romania National Olympiad, 3

Let $A,B \in \mathcal{M}_n(\mathbb{R}).$ We consider the function $f:\mathcal{M}_n(\mathbb{C}) \to \mathcal{M}_n(\mathbb{C}),$ defined by $f(Z)=AZ+B\overline{Z},$ $Z \in \mathcal{M}_n(\mathbb{C}),$ where $\overline{Z}$ is the matrix whose entries are the conjugates of the corresponding entries of $Z.$ Prove that the following statements are equivalent: $(1)$ the function $f$ is injective; $(2)$ the function $f$ is surjective; $(3)$ the matrices $A+B$ and $A-B$ are invertible.

2000 District Olympiad (Hunedoara), 1

Solve in the set of $ 2\times 2 $ integer matrices the equation $$ X^2-4\cdot X+4\cdot\left(\begin{matrix}1\quad 0\\0\quad 1\end{matrix}\right) =\left(\begin{matrix}7\quad 8\\12\quad 31\end{matrix}\right) . $$

2001 SNSB Admission, 1

Show that $ \det \left( I_n+A \right)\ge 1, $ for any $ n\times n $ antisymmetric real matrix $ A. $

2016 IMC, 5

Tags: matrix
Let $A$ be a $n\times n$ complex matrix whose eigenvalues have absolute value at most $1$. Prove that $$ \|A^n\|\le \dfrac{n}{\ln 2} \|A\|^{n-1}. $$ (Here $\|B\|=\sup\limits_{\|x\|\leq 1} \|Bx\|$ for every $n\times n$ matrix $B$ and $\|x\|=\sqrt{\sum\limits_{i=1}^n |x_i|^2}$ for every complex vector $x\in\mathbb{C}^n$.) (Proposed by Ian Morris and Fedor Petrov, St. Petersburg State University)

2019 Jozsef Wildt International Math Competition, W. 44

We consider a natural number $n$, $n \geq 2$ and the matrices \begin{tabular}{cc} $A= \begin{pmatrix} 1 & 2 & 3 & \cdots & n\\ n & 1 & 2 & \cdots & n - 1\\ n - 1 & n & 1 & \cdots & n - 2\\ \cdots & \cdots & \cdots & \cdots & \cdots\\2 & 3 & 4 & \cdots & 1 \end{pmatrix}$ \end{tabular} Show that$$\epsilon^ndet\left(I_n-A^{2n}\right)+\epsilon^{n-1}det\left(\epsilon I_n-A^{2n}\right)+\epsilon^{n-2}det\left(\epsilon^2 I_n-A^{2n}\right)+\cdots +det\left(\epsilon^n I_n-A^{2n}\right)$$ $$=n(-1)^{n-1}\left[\frac{n^n(n+1)}{2}\right]^{2n^2-4n}\left(1+(n+1)^{2n}\left(2n+(-1)^n{{2n}\choose{n}}\right)\right)$$where $\epsilon \in \mathbb{C}\backslash \mathbb{R}$, $\epsilon^{n+1}=1$

2010 National Olympiad First Round, 2

How many ordered pairs of positive integers $(x,y)$ are there such that $y^2-x^2=2y+7x+4$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 0 \qquad\textbf{(E)}\ \text{Infinitely many} $

2017 District Olympiad, 4

Tags: matrix
Let be a natural number $ n\ge 2, $ and a matrix $ A\in\mathcal{M}_n\left( \mathbb{C} \right) $ whose determinant vanishes. Show that $$ \left( A^* \right)^2 =A^*\cdot\text{tr} A^*, $$ where $ A^* $ is the adjugate of $ A. $

2019 Teodor Topan, 4

Calculate the minimum value of $ \text{tr} (A^tA) , $ where $ A $ in the cases where is a matrix of pairwise distinct nonnegative integers and: [b]a)[/b] $ \det A\equiv 1\pmod 2 $ [b]b)[/b] $ \det A=0 $ [i]Vlad Mihaly[/i]

1983 Iran MO (2nd round), 3

Find a matrix $A_{(2 \times 2)}$ for which \[ \begin{bmatrix}2 &1 \\ 3 & 2\end{bmatrix} A \begin{bmatrix}3 & 2 \\ 4 & 3\end{bmatrix} = \begin{bmatrix}1 & 2 \\ 2 & 1\end{bmatrix}.\]

2005 Brazil Undergrad MO, 6

Prove that for any natural numbers $0 \leq i_1 < i_2 < \cdots < i_k$ and $0 \leq j_1 < j_2 < \cdots < j_k$, the matrix $A = (a_{rs})_{1\leq r,s\leq k}$, $a_{rs} = {i_r + j_s\choose i_r} = {(i_r + j_s)!\over i_r!\, j_s!}$ ($1\leq r,s\leq k$) is nonsingular.

1991 Spain Mathematical Olympiad, 2

Given two distinct elements $a,b \in \{-1,0,1\}$, consider the matrix $A$ . Find a subset $S$ of the set of the rows of $A$, of minimum size, such that every other row of $A$ is a linear combination of the rows in $S$ with integer coefficients.

2003 IMC, 1

Let $A,B \in \mathbb{R}^{n\times n}$ such that $AB+B+A=0$. Prove that $AB=BA$.