Olympiad problems

2010 SEEMOUS, Problem 4

Suppose that $A$ and $B$ are $n\times n$ matrices with integer entries, and $\det B\ne0$. Prove that there exists $m\in\mathbb N$ such that the product $AB^{-1}$ can be represented as $$AB^{-1}=\sum_{k=1}^mN_k^{-1},$$where $N_k$ are $n\times n$ matrices with integer entries for all $k=1,\ldots,m$, and $N_i\ne N_j$ for $i\ne j$.

1941 Putnam, A7

Tags: linear algebra , matrix , ellipse , conic

Do either (1) or (2): (1) Prove that the determinant of the matrix $$\begin{pmatrix} 1+a^2 -b^2 -c^2 & 2(ab+c) & 2(ac-b)\\ 2(ab-c) & 1-a^2 +b^2 -c^2 & 2(bc+a)\\ 2(ac+b)& 2(bc-a) & 1-a^2 -b^2 +c^2 \end{pmatrix}$$ is given by $(1+a^2 +b^2 +c^2)^{3}$. (2) A solid is formed by rotating the first quadrant of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ around the $x$-axis. Prove that this solid can rest in stable equilibrium on its vertex if and only if $\frac{a}{b}\leq \sqrt{\frac{8}{5}}$.

2000 IMC, 3

Tags: matrix , function , vector , linear algebra

Let $A,B\in\mathbb{C}^{n\times n}$ with $\rho(AB - BA) = 1$. Show that $(AB - BA)^2 = 0$.

1954 Putnam, A1

Tags: linear algebra , matrix , permutation

Let $n$ be an odd integer greater than $1.$ Let $A$ be an $n\times n$ symmetric matrix such that each row and column consists of some permutation of the integers $1,2, \ldots, n.$ Show that each of the integers $1,2, \ldots, n$ must appear in the main diagonal of $A$.

1940 Putnam, A8

Tags: geometry , linear algebra , matrix

A triangle is bounded by the lines $a_1 x+ b_1 y +c_1=0$, $a_2 x+ b_2 y +c_2=0$ and $a_2 x+ b_2 y +c_2=0$. Show that its area, disregarding sign, is $$\frac{\Delta^{2}}{2(a_2 b_3- a_3 b_2)(a_3 b_1- a_1 b_3)(a_1 b_2- a_2 b_1)},$$ where $\Delta$ is the discriminant of the matrix $$M=\begin{pmatrix} a_1 & b_1 &c_1\\ a_2 & b_2 &c_2\\ a_3 & b_3 &c_3 \end{pmatrix}.$$

2011 Romania National Olympiad, 1

Tags: matrix , linear algebra

[color=darkred]A row of a matrix belonging to $\mathcal{M}_n(\mathbb{C})$ is said to be [i]permutable[/i] if no matter how we would permute the entries of that row, the value of the determinant doesn't change. Prove that if a matrix has two [i]permutable[/i] rows, then its determinant is equal to $0$ .[/color]

2024 District Olympiad, P1

Tags: linear algebra , matrix

Consider the matrix $X\in\mathcal{M}_2(\mathbb{C})$ which satisfies $X^{2022}=X^{2023}.$ Prove that $X^2=X^3.$

2019 VJIMC, 3

Tags: linear algebra , matrix

For an invertible $n\times n$ matrix $M$ with integer entries we define a sequence $\mathcal{S}_M=\{M_i\}_{i=0}^{\infty}$ by the recurrence $M_0=M$ ,$M_{i+1}=(M_i^T)^{-1}M_i$ for $i\geq 0$. Find the smallest integer $n\geq 2 $ for wich there exists a normal $n\times n$ matrix with integer entries such that its sequence $\mathcal{S}_M$ is not constant and has period $P=7$ i.e $M_{i+7}=M_i$. ($M^T$ means the transpose of a matrix $M$ . A square matrix is called normal if $M^T M=M M^T$ holds). [i]Proposed by Martin Niepel (Comenius University, Bratislava)..[/i]

2019 LIMIT Category C, Problem 4

Tags: linear algebra , matrix

Which of the following are true? $\textbf{(A)}~\exists A\in M_3(\mathbb R)\text{ such that }A^2=-I_3$ $\textbf{(B)}~\exists A,B\in M_3(\mathbb R)\text{ such that }AB-BA=I_3$ $\textbf{(C)}~\forall A\in M_4,\det\left(I_4+A^2\right)\ge0$ $\textbf{(D)}~\text{None of the above}$

2002 AMC 10, 20

Tags: symmetry , geometry , geometric transformation , rotation , linear algebra

Let $ a$, $ b$, and $ c$ be real numbers such that $ a \minus{} 7b \plus{} 8c \equal{} 4$ and $ 8a \plus{} 4b \minus{} c \equal{} 7$. Then $ a^2 \minus{} b^2 \plus{} c^2$ is $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$

2000 District Olympiad (Hunedoara), 1

Tags: linear algebra , matrix , algebra

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

2006 Petru Moroșan-Trident, 2

Tags: linear algebra , matrix

Let be the sequence of sets $ \left(\left\{ A\in\mathcal{M}_2\left(\mathbb{R} \right) | A^{n+1} =2007^nA\right\}\right)_{n\ge 1} . $ [b]a)[/b] Prove that each term of the above sequence hasn't a finite cardinal. [b]b)[/b] Determine the intersection of the fourth element of the above sequence with the $ 2007\text{th} $ element. [i]Gheorghe Iurea[/i] [hide=Note]Similar with [url]https://artofproblemsolving.com/community/c7h1928039p13233629[/url].[/hide]

2001 Miklós Schweitzer, 8

Tags: functional analysis , linear algebra

Let $H$ be a complex Hilbert space. The bounded linear operator $A$ is called [i]positive[/i] if $\langle Ax, x\rangle \geq 0$ for all $x\in H$. Let $\sqrt A$ be the positive square root of $A$, i.e. the uniquely determined positive operator satisfying $(\sqrt{A})^2=A$. On the set of positive operators we introduce the $$A\circ B=\sqrt A B\sqrt B$$ operation. Prove that for a given pair $A, B$ of positive operators the identity $$(A\circ B)\circ C=A\circ (B\circ C)$$ holds for all positive operator $C$ if and only if $AB=BA$.

2010 National Olympiad First Round, 2

Tags: linear algebra , matrix

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} $

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]

2005 Romania National Olympiad, 1

Tags: matrix , linear algebra

Let $n\geq 2$ a fixed integer. We shall call a $n\times n$ matrix $A$ with rational elements a [i]radical[/i] matrix if there exist an infinity of positive integers $k$, such that the equation $X^k=A$ has solutions in the set of $n\times n$ matrices with rational elements. a) Prove that if $A$ is a radical matrix then $\det A \in \{-1,0,1\}$ and there exists an infinity of radical matrices with determinant 1; b) Prove that there exist an infinity of matrices that are not radical and have determinant 0, and also an infinity of matrices that are not radical and have determinant 1. [i]After an idea of Harazi[/i]

2005 Moldova Team Selection Test, 3

Tags: linear algebra , matrix , trigonometry , combinatorics

Does there exist such a configuration of 22 circles and 22 point, that any circle contains at leats 7 points and any point belongs at least to 7 circles?

2012 Iran MO (3rd Round), 4

Tags: linear algebra , matrix , combinatorics

Prove that if $n$ is large enough, in every $n\times n$ square that a natural number is written on each one of its cells, one can find a subsquare from the main square such that the sum of the numbers is this subsquare is divisible by $1391$.

2008 Putnam, A3

Tags: least common multiple , invariant , induction , linear algebra , matrix , number theory

Start with a finite sequence $ a_1,a_2,\dots,a_n$ of positive integers. If possible, choose two indices $ j < k$ such that $ a_j$ does not divide $ a_k$ and replace $ a_j$ and $ a_k$ by $ \gcd(a_j,a_k)$ and $ \text{lcm}\,(a_j,a_k),$ respectively. Prove that if this process is repeated, it must eventually stop and the final sequence does not depend on the choices made. (Note: $ \gcd$ means greatest common divisor and lcm means least common multiple.)

1995 IMC, 7

Tags: linear algebra , matrix , vector

Let $A$ be a $3\times 3$ real matrix such that the vectors $Au$ and $u$ are orthogonal for every column vector $u\in \mathbb{R}^{3}$. Prove that: a) $A^{T}=-A$. b) there exists a vector $v \in \mathbb{R}^{3}$ such that $Au=v\times u$ for every $u\in \mathbb{R}^{3}$, where $v \times u$ denotes the vector product in $\mathbb{R}^{3}$.

2012 CIIM, Problem 1

Tags: linear algebra , matrix

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 $A_n$.

2017-IMOC, C3

Tags: linear algebra , matrix

Alice and Bob play the following game: Initially, there is a $2016\times2016$ "empty" matrix. Taking turns, with Alice playing first, each player chooses a real number and fill it into an empty entry. If the determinant of the last matrix is non-zero, then Alice wins. Otherwise, Bob wins. Who has the winning strategy?

1997 Iran MO (3rd Round), 3

Tags: algebra , linear algebra

Let $S = \{x_0, x_1,\dots , x_n\}$ be a finite set of numbers in the interval $[0, 1]$ with $x_0 = 0$ and $x_1 = 1$. We consider pairwise distances between numbers in $S$. If every distance that appears, except the distance $1$, occurs at least twice, prove that all the $x_i$ are rational.

2020 Miklós Schweitzer, 3

Tags: linear algebra , matrix , vector

An $n\times n$ matrix $A$ with integer entries is called [i]representative[/i] if, for any integer vector $\mathbf{v}$, there is a finite sequence $0=\mathbf{v}_0,\mathbf{v}_1,\dots,\mathbf{v}_{\ell}=\mathbf{v}$ of integer vectors such that for each $0\leq i <\ell$, either $\mathbf{v}_{i+1}=A\mathbf{v}_{i}$ or $\mathbf{v}_{i+1}-\mathbf{v}_i$ is an element of the standard basis (i.e. one of its entries is $1$, the rest are all equal to $0$). Show that $A$ is not representative if and only if $A^T$ has a real eigenvector with all non-negative entries and non-negative eigenvalue.

2005 Alexandru Myller, 1

Tags: linear algebra , matrix , superior algebra

Let $A,B\in M_2(\mathbb Z)$ s.t. $AB=\begin{pmatrix}1&2005\\0&1\end{pmatrix}$. Prove that there is a matrix $C\in M_2(\mathbb Z)$ s.t. $BA=C^{2005}$. [i]Dinu Serbanescu[/i]