Olympiad problems

2012 Junior Balkan Team Selection Tests - Moldova, 4

How many solutions does the system have: $ \{\begin{matrix}&(3x+2y) *(\frac{3}{x}+\frac{1}{y})=2\\ & x^2+y^2\leq 2012\\ \end{matrix} $ where $ x,y $ are non-zero integers

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.

2023 IMC, 6

Tags: combinatorics , matrix , linear algebra

Ivan writes the matrix $\begin{pmatrix} 2 & 3\\ 2 & 4\end{pmatrix}$ on the board. Then he performs the following operation on the matrix several times: [b]1.[/b] he chooses a row or column of the matrix, and [b]2.[/b] he multiplies or divides the chosen row or column entry-wise by the other row or column, respectively. Can Ivan end up with the matrix $\begin{pmatrix} 2 & 4\\ 2 & 3\end{pmatrix}$ after finitely many steps?

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 $A^2=2018 A$.

2009 Mediterranean Mathematics Olympiad, 3

Tags: combinatorics , matrix , linear algebra

Decide whether the integers $1,2,\ldots,100$ can be arranged in the cells $C(i, j)$ of a $10\times10$ matrix (where $1\le i,j\le 10$), such that the following conditions are fullfiled: i) In every row, the entries add up to the same sum $S$. ii) In every column, the entries also add up to this sum $S$. iii) For every $k = 1, 2, \ldots, 10$ the ten entries $C(i, j)$ with $i-j\equiv k\bmod{10}$ add up to $S$. [i](Proposed by Gerhard Woeginger, Austria)[/i]

2007 Nicolae Păun, 1

Tags: matrix , linear algebra

Prove that $ \exists X,Y,Z\in \mathcal{M}_n(\mathbb{C})$ such that a)$ X^2\plus{}Y^2\equal{}A$ b) $ X^3\plus{}Y^3\plus{}Z^3\equal{}A$ , where $ A\in \mathcal{M}_n(\mathbb{C})$

2011 Morocco National Olympiad, 3

Tags: linear algebra , matrix , algebra

Solve in $\mathbb{R}^{3}$ the following system \[\left\{\begin{matrix} \sqrt{x^{2}-y}=z-1\\ \sqrt{y^{2}-z}=x-1\\ \sqrt{z^{2}-x}=y-1 \end{matrix}\right.\]

1994 China Team Selection Test, 1

Tags: logarithm , algebra , inequalities , function , domain , linear algebra , matrix

Given $5n$ real numbers $r_i, s_i, t_i, u_i, v_i \geq 1 (1 \leq i \leq n)$, let $R = \frac {1}{n} \sum_{i=1}^{n} r_i$, $S = \frac {1}{n} \sum_{i=1}^{n} s_i$, $T = \frac {1}{n} \sum_{i=1}^{n} t_i$, $U = \frac {1}{n} \sum_{i=1}^{n} u_i$, $V = \frac {1}{n} \sum_{i=1}^{n} v_i$. Prove that $\prod_{i=1}^{n}\frac {r_i s_i t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \geq \left(\frac {RSTUV +1}{RSTUV - 1}\right)^n$.

2007 VJIMC, Problem 2

Tags: linear algebra , matrix

Let $A$ be a real $n\times n$ matrix satisfying $$A+A^{\text T}=I,$$where $A^{\text T}$ denotes the transpose of $A$ and $I$ the $n\times n$ identity matrix. Show that $\det A>0$.

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]

1997 IMC, 2

Tags: linear algebra , matrix

Let $M \in GL_{2n}(K)$, represented in block form as \[ M = \left[ \begin{array}{cc} A & B \\ C & D \end{array} \right] , M^{-1} = \left[ \begin{array}{cc} E & F \\ G & H \end{array} \right] \] Show that $\det M.\det H=\det A$.

2022 Miklós Schweitzer, 2

Tags: linear algebra , matrix

Original in Hungarian; translated with Google translate; polished by myself. Let $n$ be a positive integer. Suppose that the sum of the matrices $A_1, \dots, A_n\in \Bbb R^{n\times n}$ is the identity matrix, but $\sum\nolimits_{i = 1}^n\alpha_i A_i$ is singular whenever at least one of the coefficients $\alpha_i \in \Bbb R$ is zero. a) Show that $\sum\nolimits_{i = 1}^n\alpha_i A_i$ is nonsingular if $\alpha_i\neq 0$ for all $i$. b) Show that if the matrices $A_i$ are symmetric, then all of them have rank $1$.

2005 Germany Team Selection Test, 3

Tags: combinatorics , matrix , linear algebra , extremal combinatorics

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 ${\{1,2,\dots ,n\}}$.

2023 China Team Selection Test, P18

Tags: combinatorics , matrix , square grid

Find the greatest constant $\lambda$ such that for any doubly stochastic matrix of order 100, we can pick $150$ entries such that if the other $9850$ entries were replaced by $0$, the sum of entries in each row and each column is at least $\lambda$. Note: A doubly stochastic matrix of order $n$ is a $n\times n$ matrix, all entries are nonnegative reals, and the sum of entries in each row and column is equal to 1.

2009 VJIMC, Problem 3

Tags: matrix , linear algebra

Let $A$ be an $n\times n$ square matrix with integer entries. Suppose that $p^2A^{p^2}=q^2A^{q^2}+r^2I_n$ for some positive integers $p,q,r$ where $r$ is odd and $p^2=q^2+r^2$. Prove that $|\det A|=1$. (Here $I_n$ means the $n\times n$ identity matrix.)

2004 Bulgaria Team Selection Test, 3

Tags: linear algebra , combinatorics , induction , matrix

In any cell of an $n \times n$ table a number is written such that all the rows are distinct. Prove that we can remove a column such that the rows in the new table are still distinct.

2005 India IMO Training Camp, 3

Tags: linear algebra , combinatorics , matrix , probability , algebra

Consider a matrix of size $n\times n$ whose entries are real numbers of absolute value not exceeding $1$. The sum of all entries of the matrix is $0$. Let $n$ be an even positive integer. Determine the least number $C$ such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding $C$ in absolute value. [i]Proposed by Marcin Kuczma, Poland[/i]

1997 Brazil Team Selection Test, Problem 4

Tags: number theory , matrix , pigeonhole principle

Consider an $N\times N$ matrix, where $N$ is an odd positive integer, such that all its entries are $-1,0$ or $1$. Consider the sum of the numbers in every line and every column. Prove that at least two of the $2N$ sums are equal.

1995 Italy TST, 2

Tags: symmetry , matrix , geometry , geometric transformation , rectangle , linear algebra , reflection

Twenty-one rectangles of size $3\times 1$ are placed on an $8\times 8$ chessboard, leaving only one free unit square. What position can the free square lie at?

2013 Argentina Cono Sur TST, 4

Tags: number theory , linear algebra , matrix

Show that the number $\begin{matrix} \\ N= \end{matrix} \underbrace{44 \ldots 4}_{n} \underbrace{88 \ldots 8}_{n} - 1\underbrace{33 \ldots3 }_{n-1}2$ is a perfect square for all positive integers $n$.

2009 All-Russian Olympiad, 6

Tags: induction , linear algebra , topology , euler , 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$.

2018 Korea USCM, 5

Tags: matrix , linear algebra , spectrum

A real symmetric $2018\times 2018$ matrix $A=(a_{ij})$ satisfies $|a_{ij}-2018|\leq 1$ for every $1\leq i,j\leq 2018$. Denote the largest eigenvalue of $A$ by $\lambda(A)$. Find maximum and minumum value of $\lambda(A)$.

2025 VJIMC, 4

Tags: linear algebra , matrix , algebra , polynomial , trace , minimal polynomial

Let $A$ be an $n\times n$ real matrix with minimal polynomial $x^n + x - 1$. Prove that the trace of $(nA^{n-1} + I)^{-1}A^{n-2}$ is zero.

2002 District Olympiad, 3

Tags: function , matrix , linear algebra

a)Find a matrix $A\in \mathcal{M}_3(\mathbb{C})$ such that $A^2\neq O_3$ and $A^3=O_3$. b)Let $n,p\in\{2,3\}$. Prove that if there is bijective function $f:\mathcal{M}_n(\mathbb{C})\rightarrow \mathcal{M}_p(\mathbb{C})$ such that $f(XY)=f(X)f(Y),\ \forall X,Y\in \mathcal{M}_n(\mathbb{C})$, then $n=p$. [i]Ion Savu[/i]

2003 China Team Selection Test, 2

Tags: inequalities , ratio , geometry , analytic geometry , linear algebra , matrix

In triangle $ABC$, the medians and bisectors corresponding to sides $BC$, $CA$, $AB$ are $m_a$, $m_b$, $m_c$ and $w_a$, $w_b$, $w_c$ respectively. $P=w_a \cap m_b$, $Q=w_b \cap m_c$, $R=w_c \cap m_a$. Denote the areas of triangle $ABC$ and $PQR$ by $F_1$ and $F_2$ respectively. Find the least positive constant $m$ such that $\frac{F_1}{F_2}<m$ holds for any $\triangle{ABC}$.