Olympiad problems

2019 LIMIT Category C, Problem 7

Let $O(4,\mathbb Z)$ be the set of all $4\times4$ orthogonal matrices over $\mathbb Z$, i.e., $A^tA=I=AA^t$. Then $|O(4,\mathbb Z)|$ is

2019 LIMIT Category C, Problem 9

Tags: Matrices , linear algebra , matrix

$P\in A_n(\mathbb R)=\{M_{n\times n}|M^2=M\}$. Which of the following are true? $\textbf{(A)}~P^T=P,\forall P\in A_n(\mathbb R)$ $\textbf{(B)}~\exists P\ne0,P\in A_n(\mathbb R)\text{ with }\operatorname{tr}(P)=0$ $\textbf{(C)}~\exists X_{n\times r}\text{ such that }Px=X\text{ for }r=\operatorname{rank}(P)$

2022 SEEMOUS, 3

Tags: linear algebra , Matrices , hermitian , Inequality

Let $\alpha \in \mathbb{C}\setminus \{0\}$ and $A \in \mathcal{M}_n(\mathbb{C})$, $A \neq O_n$, be such that $$A^2 + (A^*)^2 = \alpha A\cdot A^*,$$ where $A^* = (\bar A)^T.$ Prove that $\alpha \in \mathbb{R}$, $|\alpha| \le 2$. and $A\cdot A^* = A^*\cdot A.$

2024 Mexican University Math Olympiad, 4

Tags: Matrices , limit , linear algebra , series

Given $ b > 0 $, consider the following matrix: \[ B = \begin{pmatrix} b & b^2 \\ b^2 & b^3 \end{pmatrix} \] Denote by $ e_i $ the top left entry of $ B^i $. Prove that the following limit exists and calculate its value: \[ \lim_{i \to \infty} \sqrt[i]{e_i}. \]

2019 District Olympiad, 3

Tags: determinan , Matrices , linear algebra

Let $n$ be an odd natural number and $A,B \in \mathcal{M}_n(\mathbb{C})$ be two matrices such that $(A-B)^2=O_n.$ Prove that $\det(AB-BA)=0.$

2018 District Olympiad, 1

Tags: Matrices , linear algebra

Show that if $n\ge 2$ is an integer, then there exist invertible matrices $A_1, A_2, \ldots, A_n \in \mathcal{M}_2(\mathbb{R})$ with non-zero entries such that: \[A_1^{-1} + A_2^{-1} + \ldots + A_n^{-1} = (A_1 + A_2 + \ldots + A_n)^{-1}.\] [hide=Edit.] The $77777^{\text{th}}$ topic in College Math :coolspeak: [/hide]

2012 Mathcenter Contest + Longlist, 1 sl8

Tags: matrix , Matrices , linear algebra , algebra

For matrices $A=[a_{ij}]_{m \times m}$ and $B=[b_{ij}]_{m \times m}$ where $A,B \in \mathbb{Z} ^{m \times m}$ let $A \equiv B \pmod{n}$ only if $a_{ij} \equiv b_{ij} \pmod{n}$ for every $i,j \in \{ 1,2,...,m \}$, that's $A-B=nZ$ for some $Z \in \mathbb{Z}^{m \times m}$. (The symbol $A \in \mathbb{Z} ^{m \times m}$ means that every element in $A$ is an integer.) Prove that for $A \in \mathbb{Z} ^{m \times m}$ there is $B \in \mathbb{Z} ^{m \times m}$ , where $AB \equiv I \pmod{n }$ only if $(\det (A),n)=1$ and find the value of $B$ in the form of $A$ where $I$ represents the dimensional identity matrix $m \times m$. [i](PP-nine)[/i]

2024 Mexican University Math Olympiad, 2

Tags: Matrices , polynomial , linear algebra , algebra

Let $ A $ and $ B $ be two square matrices with complex entries such that $ A + B = AB $, $ A = A^* $, and $ A $ has all distinct eigenvalues. Prove that there exists a polynomial $ P $ with complex coefficients such that $ P(A) = B $.

2013 Bogdan Stan, 3

Tags: eigenvalue , matries , Matrices , linear algebra

Let be four $ n\times n $ real matrices $ A,B,C,D $ having the property that $ C+D\sqrt{-1} $ is the inverse of $ A+B\sqrt{-1} . $ Show that $ \left| \det\left( A+B\sqrt{-1} \right) \right|^2\cdot\left| \det C \right| =\det A. $ [i]Vasile Pop[/i]

2004 Nicolae Coculescu, 3

Tags: Matrices , linear algebra , equations , Cayley-Hamilton

Solve in $ \mathcal{M}_2(\mathbb{R}) $ the equation $ X^3+X+2I=0. $ [i]Florian Dumitrel[/i]

2000 Romania National Olympiad, 1

Tags: Matrices , linear algebra , complex numbers

Let $ \mathcal{M} =\left\{ A\in M_2\left( \mathbb{C}\right)\big| \det\left( A-zI_2\right) =0\implies |z| < 1\right\} . $ Prove that: $$ X,Y\in\mathcal{M}\wedge X\cdot Y=Y\cdot X\implies X\cdot Y\in\mathcal{M} . $$

2019 Korea USCM, 8

Tags: vector space , Matrices , linear algebra , college contests , matrix

$M_n(\mathbb{C})$ is the vector space of all complex $n\times n$ matrices. Given a linear map $T:M_n(\mathbb{C})\to M_n(\mathbb{C})$ s.t. $\det (A)=\det(T(A))$ for every $A\in M_n(\mathbb{C})$. (1) If $T(A)$ is the zero matrix, then show that $A$ is also the zero matrix. (2) Prove that $\text{rank} (A)=\text{rank} (T(A))$ for any $A\in M_n(\mathbb{C})$.

2017 Simon Marais Mathematical Competition, A3

Tags: Matrices , linear algebra , matrix

For each positive integer $n$, let $M(n)$ be the $n\times n$ matrix whose $(i,j)$ entry is equal to $1$ if $i+1$ is divisible by $j$, and equal to $0$ otherwise. Prove that $M(n)$ is invertible if and only if $n+1$ is square-free. (An integer is [i]square-free[/i] if it is not divisible by a square of an integer larger than $1$.)

2004 Spain Mathematical Olympiad, Problem 1

Tags: math olympiad , Matrices , matrix , Arithmetic Progression , algebra unsolved

We have a set of ${221}$ real numbers whose sum is ${110721}$. It is deemed that the numbers form a rectangular table such that every row as well as the first and last columns are arithmetic progressions of more than one element. Prove that the sum of the elements in the four corners is equal to ${2004}$.

2019 Brazil Undergrad MO, 1

Tags: college contests , Matrices , linear algebra , polynomial , Brazilian Undergrad MO 2019 , Brazilian Undergrad MO

Let $ I $ and $ 0 $ be the square identity and null matrices, both of size $ 2019 $. There is a square matrix $A$ with rational entries and size $ 2019 $ such that: a) $ A ^ 3 + 6A ^ 2-2I = 0 $? b) $ A ^ 4 + 6A ^ 3-2I = 0 $?

1967 Putnam, A2

Tags: Putnam , Matrices , power series , generating functions

Define $S_0$ to be $1.$ For $n \geq 1 $, let $S_n $ be the number of $n\times n $ matrices whose elements are nonnegative integers with the property that $a_{ij}=a_{ji}$ (for $i,j=1,2,\ldots, n$) and where $\sum_{i=1}^{n} a_{ij}=1$ (for $j=1,2,\ldots, n$). Prove that a) $S_{n+1}=S_{n} +nS_{n-1}.$ b) $\sum_{n=0}^{\infty} S_{n} \frac{x^{n}}{n!} =\exp \left(x+\frac{x^{2}}{2}\right).$

2015 VJIMC, 1

Tags: Matrices , linear algebra , college contests

[b]Problem 1 [/b] Let $A$ and $B$ be two $3 \times 3$ matrices with real entries. Prove that $$ A-(A^{-1} +(B^{-1}-A)^{-1})^{-1} =ABA\ , $$ provided all the inverses appearing on the left-hand side of the equality exist.

1996 Romania National Olympiad, 4

Tags: linear algebra , Matrices

Let $A,B,C,D \in \mathcal{M}_n(\mathbb{C}),$ $A$ and $C$ invertible. Prove that if $A^k B = C^k D$ for any positive integer $k,$ then $B=D.$

1996 Romania National Olympiad, 3

Tags: Matrices , Determinants , linear algebra

Let $A, B \in M_2(\mathbb{R})$ such as $det(AB+BA)\leq 0$. Prove that $$det(A^2+B^2)\geq 0$$

2022 Romania National Olympiad, P4

Tags: linear algebra , Matrices , romania

Let $A,B\in\mathcal{M}_n(\mathbb{C})$ such that $A^2+B^2=2AB.$ Prove that for any complex number $x$\[\det(A-xI_n)=\det(B-xI_n).\][i]Mihai Opincariu and Vasile Pop[/i]

2021 Alibaba Global Math Competition, 3

Tags: Matrices , binary matrix , college contests

Given positive integers $k \ge 2$ and $m$ sufficiently large. Let $\mathcal{F}_m$ be the infinite family of all the (not necessarily square) binary matrices which contain exactly $m$ 1's. Denote by $f(m)$ the maximum integer $L$ such that for every matrix $A \in \mathcal{F}_m$, there always exists a binary matrix $B$ of the same dimension such that (1) $B$ has at least $L$ 1-entries; (2) every entry of $B$ is less or equal to the corresponding entry of $A$; (3) $B$ does not contain any $k \times k$ all-1 submatrix. Show the equality \[\lim_{m \to \infty} \frac{\ln f(m)}{\ln m}=\frac{k}{k+1}.\]

2006 Cezar Ivănescu, 2

Tags: group theory , abstract algebra , Matrices , isomorphism , group morphism

Prove that the set $ \left\{ \left. \begin{pmatrix} \frac{1-2x^3}{3x^2} & \frac{1+x^3}{3x^2} & \frac{1+x^3}{3x^2} \\ \frac{1+x^3}{3x^2} & \frac{1-2x^3}{3x^2} & \frac{1+x^3}{3x^2} \\ \frac{1+x^3}{3x^2} & \frac{1+x^3}{3x^2} & \frac{1-2x^3}{3x^2}\end{pmatrix}\right| x\in\mathbb{R}^{*} \right\} $ along with the usual multiplication of matrices form a group, determine an isomorphism between this group and the group of multiplicative real numbers.

2024 IMC, 7

Tags: linear algebra , determinant , Matrices

Let $n$ be a positive integer. Suppose that $A$ and $B$ are invertible $n \times n$ matrices with complex entries such that $A+B=I$ (where $I$ is the identity matrix) and \[(A^2+B^2)(A^4+B^4)=A^5+B^5.\] Find all possible values of $\det(AB)$ for the given $n$.