Found problems: 638
2009 Stars Of Mathematics, 5
The cells of a $(n^2-n+1)\times(n^2-n+1)$ matrix are coloured using $n$ colours. A colour is called [i]dominant[/i] on a row (or a column) if there are at least $n$ cells of this colour on that row (or column). A cell is called [i]extremal[/i] if its colour is [i]dominant [/i] both on its row, and its column. Find all $n \ge 2$ for which there is a colouring with no [i]extremal [/i] cells.
Iurie Boreico (Moldova)
1940 Putnam, A8
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
$$M=\begin{pmatrix}
a_1 & b_1 &c_1\\
a_2 & b_2 &c_2\\
a_3 & b_3 &c_3
\end{pmatrix}.$$
2013 IMC, 1
Let $\displaystyle{A}$ and $\displaystyle{B}$ be real symmetric matrixes with all eigenvalues strictly greater than $\displaystyle{1}$. Let $\displaystyle{\lambda }$ be a real eigenvalue of matrix $\displaystyle{{\rm A}{\rm B}}$. Prove that $\displaystyle{\left| \lambda \right| > 1}$.
[i]Proposed by Pavel Kozhevnikov, MIPT, Moscow.[/i]
2008 Junior Balkan MO, 4
A $ 4\times 4$ table is divided into $ 16$ white unit square cells. Two cells are called neighbors if they share a common side. A [i]move[/i] consists in choosing a cell and the colors of neighbors from white to black or from black to white. After exactly $ n$ moves all the $ 16$ cells were black. Find all possible values of $ n$.
2012 VJIMC, Problem 2
Determine all $2\times2$ integer matrices $A$ having the following properties:
$1.$ the entries of $A$ are (positive) prime numbers,
$2.$ there exists a $2\times2$ integer matrix $B$ such that $A=B^2$ and the determinant of $B$ is the square of a prime number.
1998 IMO Shortlist, 1
A rectangular array of numbers is given. In each row and each column, the sum of all numbers is an integer. Prove that each nonintegral number $x$ in the array can be changed into either $\lceil x\rceil $ or $\lfloor x\rfloor $ so that the row-sums and column-sums remain unchanged. (Note that $\lceil x\rceil $ is the least integer greater than or equal to $x$, while $\lfloor x\rfloor $ is the greatest integer less than or equal to $x$.)
2006 Petru Moroșan-Trident, 2
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]
2020 SEEMOUS, Problem 3
Let $n$ be a positive integer, $k\in \mathbb{C}$ and $A\in \mathcal{M}_n(\mathbb{C})$ such that $\text{Tr } A\neq 0$ and $$\text{rank } A +\text{rank } ((\text{Tr } A) \cdot I_n - kA) =n.$$
Find $\text{rank } A$.
1999 IMO Shortlist, 5
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.
1987 Greece National Olympiad, 2
Let $A=(\alpha_{ij})$ be a $m\,x\,n$ matric and $B=(\beta_{kl})$ be a $n\,x\, m$ matric with $m>n$ . Prove that $D(A\cdot B)=0$.
1996 IMC, 1
Let $A=(a_{ij})\in M_{(n+1)\times (n+1)}(\mathbb{R})$ with $a_{ij}=a+|i-j|d$, where $a$ and $d$ are fixed real numbers.
Calculate $\det(A)$.
2021 IMC, 5
Let $A$ be a real $n \times n$ matrix and suppose that for every positive integer $m$ there exists a real symmetric matrix $B$ such that
$$2021B = A^m+B^2.$$
Prove that $|\text{det} A| \leq 1$.
1997 IMC, 2
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 $\det M.\det H=\det A$.
1987 AMC 12/AHSME, 25
$ABC$ is a triangle: $A=(0,0)$, $B=(36,15)$ and both the coordinates of $C$ are integers. What is the minimum area $\triangle ABC$ can have?
$ \textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \frac{3}{2} \qquad\textbf{(D)}\ \frac{13}{2} \qquad\textbf{(E)}\ \text{there is no minimum} $
2004 VTRMC, Problem 1
Let $I$ denote the $2\times2$ identity matrix $\begin{pmatrix}1&0\\0&1\end{pmatrix}$ and let
$$M=\begin{pmatrix}I&A\\B&C\end{pmatrix},\enspace N=\begin{pmatrix}I&B\\A&C\end{pmatrix}$$where $A,B,C$ are arbitrary $2\times2$ matrices which entries in $\mathbb R$, the real numbers. Thus $M$ and $N$ are $4\times4$ matrices with entries in $\mathbb R$. Is it true that $M$ is invertible (i.e. there is a $4\times4$ matrix $X$ such that $MX=XM=I$) implies $N$ is invertible? Justify your answer.
2005 Taiwan TST Round 1, 3
$n$ teams take part in a tournament, in which every two teams compete exactly once, and that no draws are possible. It is known that for any two teams, there exists another team which defeated both of the two teams. Find all $n$ for which this is possible.
2004 Bulgaria Team Selection Test, 3
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.
2004 Italy TST, 3
Given real numbers $x_i,y_i (i=1,2,\ldots ,n)$, let $A$ be the $n\times n$ matrix given by $a_{ij}=1$ if $x_i\ge y_j$ and $a_{ij}=0$ otherwise. Suppose $B$ is a $n\times n$ matrix whose entries are $0$ and $1$ such that the sum of entries in any row or column of $B$ equals the sum of entries in the corresponding row or column of $A$. Prove that $B=A$.
2003 Alexandru Myller, 1
Let be the sequence of sets $ \left(\left\{ A\in\mathcal{M}_2\left(\mathbb{R} \right) | A^{n+1} =2003^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 third element of the above sequence with the $ 2003\text{rd} $ element.
[i]Gheorghe Iurea[/i]
[hide=Note]Similar with [url]https://artofproblemsolving.com/community/c7h1943241p13387495[/url].[/hide]
2022 CIIM, 2
Let $v \in \mathbb{R}^2$ a vector of length 1 and $A$ a $2 \times 2$ matrix with real entries such that:
(i) The vectors $A v, A^2 v$ y $A^3 v$ are also of length 1.
(ii) The vector $A^2 v$ isn't equal to $\pm v$ nor to $\pm A v$.
Prove that $A^t A=I_2$.
2003 VJIMC, Problem 1
Two real square matrices $A$ and $B$ satisfy the conditions $A^{2002}=B^{2003}=I$ and $AB=BA$. Prove that $A+B+I$ is invertible. (The symbol $I$ denotes the identity matrix.)
2018 Ramnicean Hope, 1
Let be a natural number $ n\ge 2, $ the real numbers $ a_1,a_2,\ldots ,a_n,b_1,b_2,\ldots, b_n, $ and the matrix defined as
$$ A=\left( a_i+b_j \right)_{1\le j\le n}^{1\le i\le n} . $$
[b]a)[/b] Show that $ n=2 $ if $ A $ is invertible.
[b]b)[/b] Prove that the pair of numbers $ a_1,a_2 $ and $ b_1,b_2 $ are both consecutive (not necessarily in this order), if $ A $ is an invertible matrix of integers whose inverse is a matrix of integers.
[i]Costică Ambrinoc[/i]
2009 Hong Kong TST, 2
Find the total number of solutions to the following system of equations:
$ \{\begin{array}{l} a^2 + bc\equiv a \pmod{37} \\
b(a + d)\equiv b \pmod{37} \\
c(a + d)\equiv c \pmod{37} \\
bc + d^2\equiv d \pmod{37} \\
ad - bc\equiv 1 \pmod{37} \end{array}$
1971 IMO Shortlist, 11
The matrix
\[A=\begin{pmatrix} a_{11} & \ldots & a_{1n} \\ \vdots & \ldots & \vdots \\ a_{n1} & \ldots & a_{nn} \end{pmatrix}\]
satisfies the inequality $\sum_{j=1}^n |a_{j1}x_1 + \cdots+ a_{jn}x_n| \leq M$ for each choice of numbers $x_i$ equal to $\pm 1$. Show that
\[|a_{11} + a_{22} + \cdots+ a_{nn}| \leq M.\]
1991 Arnold's Trivium, 90
Calculate the sum of matrix commutators $[A, [B, C]] + [B, [C, A]] + [C, [A, B]]$, where $[A, B] = AB-BA$