Found problems: 823
2007 IberoAmerican Olympiad For University Students, 1
For each pair of integers $(i,k)$ such that $1\le i\le k$, the linear transformation $P_{i,k}:\mathbb{R}^k\to\mathbb{R}^k$ is defined as:
$P_{i,k}(a_1,\cdots,a_{i-1},a_i,a_{i+1},\cdots,a_k)=(a_1,\cdots,a_{i-1},0,a_{i+1},\cdots,a_k)$
Prove that for all $n\ge2$ and for every set of $n-1$ linearly independent vectors $v_1,\cdots,v_{n-1}$ in $\mathbb{R}^n$, there is an integer $k$ such that $1\le k\le n$ and such that the vectors $P_{k,n}(v_1),\cdots,P_{k,n}(v_{n-1})$ are linearly independent.
2012 China National Olympiad, 2
Let $p$ be a prime. We arrange the numbers in ${\{1,2,\ldots ,p^2} \}$ as a $p \times p$ matrix $A = ( a_{ij} )$. Next we can select any row or column and add $1$ to every number in it, or subtract $1$ from every number in it. We call the arrangement [i]good[/i] if we can change every number of the matrix to $0$ in a finite number of such moves. How many good arrangements are there?
2006 China Team Selection Test, 3
Let $a_{i}$ and $b_{i}$ ($i=1,2, \cdots, n$) be rational numbers such that for any real number $x$ there is:
\[x^{2}+x+4=\sum_{i=1}^{n}(a_{i}x+b)^{2}\]
Find the least possible value of $n$.
2024 SEEMOUS, P2
Let $A,B\in\mathcal{M}_n(\mathbb{R})$ two real, symmetric matrices with nonnegative eigenvalues. Prove that $A^3+B^3=(A+B)^3$ if and only if $AB=O_n$.
1972 Spain Mathematical Olympiad, 1
Let $K$ be a ring with unit and $M$ the set of $2 \times 2$ matrices constituted with elements of $K$. An addition and a multiplication are defined in $M$ in the usual way between arrays. It is requested to:
a) Check that $M$ is a ring with unit and not commutative with respect to the laws of defined composition.
b) Check that if $K$ is a commutative field, the elements of$ M$ that have inverse they are characterized by the condition $ad - bc \ne 0$.
c) Prove that the subset of $M$ formed by the elements that have inverse is a multiplicative group.
2008 Irish Math Olympiad, 4
Given $ k \in [0,1,2,3]$ and a positive integer $ n$, let $ f_k(n)$ be the number of sequences $ x_1,...,x_n,$ where $ x_i \in [\minus{}1,0,1]$ for $ i\equal{}1,...,n,$ and
$ x_1\plus{}...\plus{}x_n \equiv k$ mod 4
a) Prove that $ f_1(n) \equal{} f_3(n)$ for all positive integers $ n$.
(b) Prove that
$ f_0(n) \equal{} [{3^n \plus{} 2 \plus{} [\minus{}1]^n}] / 4$
for all positive integers $ n$.
1959 Putnam, B4
Given the following matrix
$$\begin{pmatrix}
11& 17 & 25& 19& 16\\
24 &10 &13 & 15&3\\
12 &5 &14& 2&18\\
23 &4 &1 &8 &22 \\
6&20&7 &21&9
\end{pmatrix},$$
choose five of these elements, no two from the same row or column, in such a way that the minimum of these elements is as large as possible.
2006 IMC, 4
Let $v_{0}$ be the zero ector and let $v_{1},...,v_{n+1}\in\mathbb{R}^{n}$ such that the Euclidian norm $|v_{i}-v_{j}|$ is rational for all $0\le i,j\le n+1$. Prove that $v_{1},...,v_{n+1}$ are linearly dependent over the rationals.
2004 IMO Shortlist, 6
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 ${\{1,2,\dots ,n\}}$.
2016 Korea USCM, 3
Given positive integers $m,n$ and a $m\times n$ matrix $A$ with real entries.
(1) Show that matrices $X = I_m + AA^T$ and $Y = I_n + A^T A$ are invertible. ($I_l$ is the $l\times l$ unit matrix.)
(2) Evaluate the value of $\text{tr}(X^{-1}) - \text{tr}(Y^{-1})$.
2001 SNSB Admission, 6
There are $ n\ge 1 $ ordered bulbs controlled by $ n $ ordered switches such that the $ k\text{-th} $ switch controls the $ k\text{-th} $ bulb and also the $ j\text{-th} $ bulb if and only if the $ j\text{-th} $ switch controls the $ k\text{-th} $ bulb, for any $ 1\le k,j\le n. $ If all bulbs are off, show that it can be chosen some switches such that, if pushed simmultaneously, the bulbs turn all on.
2014 Contests, 2
Let $A$ be the $n\times n$ matrix whose entry in the $i$-th row and $j$-th column is \[\frac1{\min(i,j)}\] for $1\le i,j\le n.$ Compute $\det(A).$
1999 Miklós Schweitzer, 5
Let $\alpha>-2$ , $n\in \mathbb{N}$ and $y_1,\cdots,y_n$ be the solutions to the system of equations:
$\sum_{j=1}^n \frac{y_j}{j+k+\alpha}= \frac{1}{n+1+k+\alpha}$ , $k=1,\cdots,n$
Prove that $y_{j-1}y_{j+1}\leq y_j^2 \,\forall 1<j<n$
1997 Miklós Schweitzer, 4
An elementary change in a 0-1 matrix is a change in an element and with it all its horizontal, vertical, and diagonal neighbors (0 to 1 or 1 to 0). Can any 1791 x 1791 0-1 matrix be transformed into a zero matrix with elementary changes?
2010 Contests, 4
the code system of a new 'MO lock' is a regular $n$-gon,each vertex labelled a number $0$ or $1$ and coloured red or blue.it is known that for any two adjacent vertices,either their numbers or colours coincide.
find the number of all possible codes(in terms of $n$).
2011 Bogdan Stan, 2
Let be a natural number $ n\ge 2. $ Prove that there exist exactly two subsets of the set $ \left\{ \left.\left(\begin{matrix} a& b\\-b& a \end{matrix}\right)\right| a,b\in\mathbb{R} \right\} $ that are closed under multiplication and their cardinal is $ n. $
[i]Marcel Tena[/i]
2006 Harvard-MIT Mathematics Tournament, 8
In how many ways can we enter numbers from the set $\{1,2,3,4\}$ into a $4\times 4$ array so that all of the following conditions hold?
(a) Each row contains all four numbers.
(b) Each column contains all four numbers.
(c) Each "quadrant" contains all four numbers. (The quadrants are the four corner $2\times 2$ squares.)
2013 VTRMC, Problem 6
Let
\begin{align*}X&=\begin{pmatrix}7&8&9\\8&-9&-7\\-7&-7&9\end{pmatrix}\\Y&=\begin{pmatrix}9&8&-9\\8&-7&7\\7&9&8\end{pmatrix}.\end{align*}Let $A=Y^{-1}X$ and let $B$ be the inverse of $X^{-1}+A^{-1}$. Find a matrix $M$ such that $M^2=XY-BY$ (you may assume that $A$ and $X^{-1}+A^{-1}$ are invertible).
2019 Korea USCM, 1
$A = \begin{pmatrix} 2019 & 2020 & 2021 \\ 2020 & 2021 & 2022 \\ 2021 & 2022 & 2023 \end{pmatrix}$. Find $\text{rank}(A)$.
2024 District Olympiad, P1
Consider the matrix $X\in\mathcal{M}_2(\mathbb{C})$ which satisfies $X^{2022}=X^{2023}.$ Prove that $X^2=X^3.$
2021 IMC, 1
Let $A$ be a real $n\times n$ matrix such that $A^3=0$
a) prove that there is unique real $n\times n$ matrix $X$ that satisfied the equation
$X+AX+XA^2=A$
b) Express $X$ in terms of $A$
2009 AMC 12/AHSME, 9
Triangle $ ABC$ has vertices $ A\equal{}(3,0)$, $ B\equal{}(0,3)$, and $ C$, where $ C$ is on the line $ x\plus{}y\equal{}7$. What is the area of $ \triangle ABC$?
$ \textbf{(A)}\ 6\qquad
\textbf{(B)}\ 8\qquad
\textbf{(C)}\ 10\qquad
\textbf{(D)}\ 12\qquad
\textbf{(E)}\ 14$
2013 Singapore Senior Math Olympiad, 4
In the following $6\times 6$ matrix, one can choose any $k\times k$ submatrix, with $1<k\leq6 $ and add $1$ to all its entries. Is it possible to perform the operation a finite number of times so that all the entries in the $6\times 6$ matrix are multiples of $3$?
$ \begin{pmatrix}
2 & 0 & 1 & 0 & 2 & 0 \\
0 & 2 & 0 & 1 & 2 & 0 \\
1 & 0 & 2 & 0 & 2 & 0 \\
0 & 1 & 0 & 2 & 2 & 0 \\
1 & 1 & 1 & 1 & 2 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} $
Note: A $p\times q$ submatrix of a $m\times n$ matrix (with $p\leq m$, $q\leq n$) is a $p\times q$ matrix formed by taking a block of the entries of this size from the original matrix.
2011 N.N. Mihăileanu Individual, 1
Let be a natural number $ n\ge 2, $ two complex numbers $ p,q, $ and four matrices $ A,B,C,D\in\mathcal{M}_n(\mathbb{C}) $ such that $ A+B=C+D=pI,AB+CD=qI $ and $ ABCD=0. $ Show that $ BCDA=0. $
[i]Marius Cavachi[/i]
1997 Brazil Team Selection Test, Problem 3
Find all positive integers $x>1, y$ and primes $p,q$ such that $p^{x}=2^{y}+q^{x}$