Olympiad problems

2000 Italy TST, 2

Let $ ABC$ be an isosceles right triangle and $M$ be the midpoint of its hypotenuse $AB$. Points $D$ and $E$ are taken on the legs $AC$ and $BC$ respectively such that $AD=2DC$ and $BE=2EC$. Lines $AE$ and $DM$ intersect at $F$. Show that $FC$ bisects the $\angle DFE$.

2025 District Olympiad, P4

Tags: linear algebra , matrix

Find all triplets of matrices $A,B,C\in\mathcal{M}_2(\mathbb{R})$ which satisfy \begin{align*} A=BC-CB \\ B=CA-AC \\ C=AB-BA \end{align*} [i]Proposed by David Anghel[/i]

1994 IMO Shortlist, 2

Tags: linear algebra , matrix , algebra , system of equations

In a certain city, age is reckoned in terms of real numbers rather than integers. Every two citizens $x$ and $x'$ either know each other or do not know each other. Moreover, if they do not, then there exists a chain of citizens $x = x_0, x_1, \ldots, x_n = x'$ for some integer $n \geq 2$ such that $ x_{i-1}$ and $x_i$ know each other. In a census, all male citizens declare their ages, and there is at least one male citizen. Each female citizen provides only the information that her age is the average of the ages of all the citizens she knows. Prove that this is enough to determine uniquely the ages of all the female citizens.

2006 Moldova National Olympiad, 11.6

Tags: limit , linear algebra , matrix , vector , algebra

Sequences $(x_n)_{n\ge1}$, $(y_n)_{n\ge1}$ satisfy the relations $x_n=4x_{n-1}+3y_{n-1}$ and $y_n=2x_{n-1}+3y_{n-1}$ for $n\ge1$. If $x_1=y_1=5$ find $x_n$ and $y_n$. Calculate $\lim_{n\rightarrow\infty}\frac{x_n}{y_n}$.

2007 Purple Comet Problems, 15

Tags: linear algebra , matrix , group theory , abstract algebra , least common multiple

The alphabet in its natural order $\text{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ is $T_0$. We apply a permutation to $T_0$ to get $T_1$ which is $\text{JQOWIPANTZRCVMYEGSHUFDKBLX}$. If we apply the same permutation to $T_1$, we get $T_2$ which is $\text{ZGYKTEJMUXSODVLIAHNFPWRQCB}$. We continually apply this permutation to each $T_m$ to get $T_{m+1}$. Find the smallest positive integer $n$ so that $T_n=T_0$.

2006 District Olympiad, 2

Tags: matrix , inequalities , algebra , polynomial , abstract algebra , linear algebra

Let $n,p \geq 2$ be two integers and $A$ an $n\times n$ matrix with real elements such that $A^{p+1} = A$. a) Prove that $\textrm{rank} \left( A \right) + \textrm{rank} \left( I_n - A^p \right) = n$. b) Prove that if $p$ is prime then \[ \textrm{rank} \left( I_n - A \right) = \textrm{rank} \left( I_n - A^2 \right) = \ldots = \textrm{rank} \left( I_n - A^{p-1} \right) . \]

1997 Romania National Olympiad, 2

Tags: linear algebra , matrix

Let $A$ be a square matrix of odd order (at least $3$) whose entries are odd integers. Prove that if $A$ is invertible, then it is not possible for all the minors of the entries of a row to have equal absolute values.

2008 SEEMOUS, Problem 3

Tags: inequalities , functional equation , fe , matrix , linear algebra

Let $\mathcal M_n(\mathbb R)$ denote the set of all real $n\times n$ matrices. Find all surjective functions $f:\mathcal M_n(\mathbb R)\to\{0,1,\ldots,n\}$ which satisfy $$f(XY)\le\min\{f(X),f(Y)\}$$for all $X,Y\in\mathcal M_n(\mathbb R)$.

2018 Korea USCM, 2

Tags: linear algebra , rank , trace , matrix

Suppose a $n\times n$ real matrix $A$ satisfies $\text{tr}(A)=2018$, $\text{rank}(A)=1$. Prove that $A^2=2018 A$.

2001 India Regional Mathematical Olympiad, 4

Tags: linear algebra , matrix

Consider an $n \times n$ array of numbers $a_{ij}$ (standard notation). Suppose each row consists of the $n$ numbers $1,2,\ldots n$ in some order and $a_{ij} = a_{ji}$ for $i , j = 1,2, \ldots n$. If $n$ is odd, prove that the numbers $a_{11}, a_{22} , \ldots a_{nn}$ are $1,2,3, \ldots n$ in some order.

2014 IMC, 1

Tags: linear algebra , matrix

Determine all pairs $(a, b)$ of real numbers for which there exists a unique symmetric $2\times 2$ matrix $M$ with real entries satisfying $\mathrm{trace}(M)=a$ and $\mathrm{det}(M)=b$. (Proposed by Stephan Wagner, Stellenbosch University)

2022 District Olympiad, P4

Tags: matrix , rank

Let $A\in\mathcal{M}_n(\mathbb{C})$ where $n\geq 2.$ Prove that if $m=|\{\text{rank}(A^k)-\text{rank}(A^{k+1})":k\in\mathbb{N}^*\}|$ then $n+1\geq m(m+1)/2.$

2001 IMC, 5

Tags: linear algebra , matrix

Let $A$ be an $n\times n$ complex matrix such that $A \ne \lambda I_{n}$ for all $\lambda \in \mathbb{C}$. Prove that $A$ is similar to a matrix having at most one non-zero entry on the maindiagonal.

2003 Miklós Schweitzer, 2

Tags: linear algebra , matrix , vector

Let $p$ be a prime and let $M$ be an $n\times m$ matrix with integer entries such that $Mv\not\equiv 0\pmod{p}$ for any column vector $v\neq 0$ whose entries are $0$ are $1$. Show that there exists a row vector $x$ with integer entries such that no entry of $xM$ is $0\pmod{p}$. (translated by L. Erdős)

2014 CHMMC (Fall), 2

Tags: matrix , matrices , linear algebra

A matrix $\begin{bmatrix} x & y \\ z & w \end{bmatrix}$ has square root $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ if $$\begin{bmatrix} a & b \\ c & d \end{bmatrix}^2 = \begin{bmatrix} a^2 + bc &ab + bd \\ ac + cd & bc + d^2 \end{bmatrix} = \begin{bmatrix} x & y \\ z & w \end{bmatrix}$$ Determine how many square roots the matrix $\begin{bmatrix} 2 & 2 \\ 3 & 4 \end{bmatrix}$ has (complex coefficients are allowed).

2017 India IMO Training Camp, 3

Tags: linear algebra , matrix , combinatorics

Let $n \ge 1$ be a positive integer. An $n \times n$ matrix is called [i]good[/i] if each entry is a non-negative integer, the sum of entries in each row and each column is equal. A [i]permutation[/i] matrix is an $n \times n$ matrix consisting of $n$ ones and $n(n-1)$ zeroes such that each row and each column has exactly one non-zero entry. Prove that any [i]good[/i] matrix is a sum of finitely many [i]permutation[/i] matrices.

2009 All-Russian Olympiad, 6

Tags: topology , induction , euler , linear algebra , matrix , invariant , graph theory

Given a finite tree $ T$ and isomorphism $ f: T\rightarrow T$. Prove that either there exist a vertex $ a$ such that $ f(a)\equal{}a$ or there exist two neighbor vertices $ a$, $ b$ such that $ f(a)\equal{}b$, $ f(b)\equal{}a$.

2001 IMC, 1

Tags: linear algebra , matrix

Let $ n$ be a positive integer. Consider an $ n\times n$ matrix with entries $ 1,2,...,n^2$ written in order, starting at the top left and moving along each row in turn left-to-right. (e.g. for $ n \equal{} 3$ we get $ \left[\begin{array}{ccc}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right]$) We choose $ n$ entries of the matrix such that exactly one entry is chosen in each row and each column. What are the possible values of the sum of the selected entries?

2004 IMC, 4

Tags: linear algebra , matrix , algebra , polynomial , vector

For $n\geq 1$ let $M$ be an $n\times n$ complex array with distinct eigenvalues $\lambda_1,\lambda_2,\ldots,\lambda_k$, with multiplicities $m_1,m_2,\ldots,m_k$ respectively. Consider the linear operator $L_M$ defined by $L_MX=MX+XM^T$, for any complex $n\times n$ array $X$. Find its eigenvalues and their multiplicities. ($M^T$ denotes the transpose matrix of $M$).

2005 iTest, 20

Tags: linear algebra , matrix

If $A$ is the $3\times 3$ square matrix $\begin{bmatrix} 5 & 3 & 8\\ 2 & 2 & 5\\ 3 & 5 & 1 \end{bmatrix}$ and $B$ is the $4\times 4$ square matrix $\begin{bmatrix} 32 & 2 & 4 & 3 \\ 3 & 4 & 8 & 3 \\ 11 & 3 & 6 & 1 \\ 5 & 5 & 10 & 1 \end{bmatrix} $ find the sum of the determinants of $A$ and $B$.

2000 Irish Math Olympiad, 5

Tags: parabola , analytic geometry , linear algebra , matrix , algebra , conic

Consider all parabolas of the form $ y\equal{}x^2\plus{}2px\plus{}q$ for $ p,q \in \mathbb{R}$ which intersect the coordinate axes in three distinct points. For such $ p,q$, denote by $ C_{p,q}$ the circle through these three intersection points. Prove that all circles $ C_{p,q}$ have a point in common.

2011 USAMO, 6

Tags: linear algebra , matrix , inequalities , function

Let $A$ be a set with $|A|=225$, meaning that $A$ has 225 elements. Suppose further that there are eleven subsets $A_1, \ldots, A_{11}$ of $A$ such that $|A_i|=45$ for $1\leq i\leq11$ and $|A_i\cap A_j|=9$ for $1\leq i<j\leq11$. Prove that $|A_1\cup A_2\cup\ldots\cup A_{11}|\geq 165$, and give an example for which equality holds.

2014 Romania National Olympiad, 4

Tags: linear algebra , matrix

Let $ A\in\mathcal{M}_4\left(\mathbb{R}\right) $ be an invertible matrix whose trace is equal to the trace of its adjugate, which is nonzero. Show that $ A^2+I $ is singular if and only if there exists a nonzero matrix in $ \mathcal{M}_4\left( \mathbb{R} \right) $ that anti-commutes with it.

2010 Czech-Polish-Slovak Match, 3

Tags: geometry , rectangle , calculus , linear algebra , matrix , analytic geometry , number theory

Let $p$ be a prime number. Prove that from a $p^2\times p^2$ array of squares, we can select $p^3$ of the squares such that the centers of any four of the selected squares are not the vertices of a rectangle with sides parallel to the edges of the array.

2022 OMpD, 2

Tags: linear algebra , algebra , matrix , matrix algebra , trace

Let $p \geq 3$ be a prime number and let $A$ be a matrix of order $p$ with complex entries. Assume that $\text{Tr}(A) = 0$ and $\det(A - I_p) \neq 0$. Prove that $A^p \neq I_p$. Note: $\text{Tr}(A)$ is the sum of the main diagonal elements of $A$ and $I_p$ is the identity matrix of order $p$.