Found problems: 823
2024 CIIM, 2
Let $n$ be a positive integer, and let $M_n$ be the set of invertible matrices with integer entries and size $n \times n$.
(a) Find the largest possible value of $n$ such that there exists a symmetric matrix $A \in M_n$ satisfying
\[
\det(A^{20} + A^{24}) < 2024.
\]
(b) Prove that for every $n$, there exists a matrix $B \in M_n$ such that
\[
\det(B^{20} + B^{24}) < 2024.
\]
2004 Romania National Olympiad, 2
Let $n \in \mathbb N$, $n \geq 2$.
(a) Give an example of two matrices $A,B \in \mathcal M_n \left( \mathbb C \right)$ such that \[ \textrm{rank} \left( AB \right) - \textrm{rank} \left( BA \right) = \left\lfloor \frac{n}{2} \right\rfloor . \]
(b) Prove that for all matrices $X,Y \in \mathcal M_n \left( \mathbb C \right)$ we have \[ \textrm{rank} \left( XY \right) - \textrm{rank} \left( YX \right) \leq \left\lfloor \frac{n}{2} \right\rfloor . \]
[i]Ion Savu[/i]
2021 Brazil Undergrad MO, Problem 1
Consider the matrices like
$$M=
\left(
\begin{array}{ccc}
a & b & c \\
c & a & b \\
b & c & a
\end{array}
\right)$$
such that $det(M) = 1$.
Show that
a) There are infinitely many matrices like above with $a,b,c \in \mathbb{Q}$
b) There are finitely many matrices like above with $a,b,c \in \mathbb{Z}$
2017 India IMO Training Camp, 3
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.
2004 IMO Shortlist, 4
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]
2009 District Olympiad, 1
Let $A,B,C\in \mathcal{M}_3(\mathbb{R})$ such that $\det A=\det B=\det C$ and $\det(A+iB)=\det(C+iA)$. Prove that $\det (A+B)=\det (C+A)$.
2014 Contests, 3
Prove that there exists an infinite set of points \[ \dots, \; P_{-3}, \; P_{-2},\; P_{-1},\; P_0,\; P_1,\; P_2,\; P_3,\; \dots \] in the plane with the following property: For any three distinct integers $a,b,$ and $c$, points $P_a$, $P_b$, and $P_c$ are collinear if and only if $a+b+c=2014$.
2000 Romania National Olympiad, 1
Let $ \mathcal{M} =\left\{ A\in M_2\left( \mathbb{C}\right)\big| \det\left( A-zI_2\right) =0\implies |z| < 1\right\} . $ Prove that:
$$ X,Y\in\mathcal{M}\wedge X\cdot Y=Y\cdot X\implies X\cdot Y\in\mathcal{M} . $$
1976 USAMO, 5
If $ P(x),Q(x),R(x)$, and $ S(x)$ are all polynomials such that \[ P(x^5)\plus{}xQ(x^5)\plus{}x^2R(x^5)\equal{}(x^4\plus{}x^3\plus{}x^2\plus{}x\plus{}1)S(x),\] prove that $ x\minus{}1$ is a factor of $ P(x)$.
1954 Putnam, A1
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 $A$.
2002 India IMO Training Camp, 11
Let $ABC$ be a triangle and $P$ an exterior point in the plane of the triangle. Suppose the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ (or extensions thereof) in $D$, $E$, $F$, respectively. Suppose further that the areas of triangles $PBD$, $PCE$, $PAF$ are all equal. Prove that each of these areas is equal to the area of triangle $ABC$ itself.
2007 Turkey Team Selection Test, 3
We write $1$ or $-1$ on each unit square of a $2007 \times 2007$ board. Find the number of writings such that for every square on the board the absolute value of the sum of numbers on the square is less then or equal to $1$.
2010 Singapore MO Open, 4
Let $n$ be a positive integer. Find the smallest positive integer $k$ with the property that for any colouring nof the squares of a $2n$ by $k$ chessboard with $n$ colours, there are $2$ columns and $2$ rows such that the $4$ squares in their intersections have the same colour.
2005 Romania National Olympiad, 3
Let $X_1,X_2,\ldots,X_m$ a numbering of the $m=2^n-1$ non-empty subsets of the set $\{1,2,\ldots,n\}$, $n\geq 2$. We consider the matrix $(a_{ij})_{1\leq i,j\leq m}$, where $a_{ij}=0$, if $X_i \cap X_j = \emptyset$, and $a_{ij}=1$ otherwise. Prove that the determinant $d$ of this matrix does not depend on the way the numbering was done and compute $d$.
2012 Pre-Preparation Course Examination, 1
Suppose that $W,W_1$ and $W_2$ are subspaces of a vector space $V$ such that $V=W_1\oplus W_2$. Under what conditions we have
$W=(W\cap W_1)\oplus(W\cap W_2)$?
2002 Romania National Olympiad, 3
Let $A\in M_4(C)$ be a non-zero matrix.
$a)$ If $\text{rank}(A)=r<4$, prove the existence of two invertible matrices $U,V\in M_4(C)$, such that:
\[UAV=\begin{pmatrix}I_r&0\\0&0\end{pmatrix}\]
where $I_r$ is the $r$-unit matrix.
$b)$ Show that if $A$ and $A^2$ have the same rank $k$, then the matrix $A^n$ has rank $k$, for any $n\ge 3$.
2009 VTRMC, Problem 5
Suppose $A,B\in M_3(\mathbb C)$, $B\ne0$, and $AB=0$. Prove that there exists $D\in M_3(\mathbb C)$ with $D\ne0$ such that $AD=DA=0$.
2008 Teodor Topan, 1
Solve in $ M_2(\mathbb{C})$ the equation $ X^2\equal{}\left(
\begin{array}{cc}
1 & 2 \\
3 & 6 \end{array}
\right)$
2007 IMS, 4
Prove that: \[\det(A)=\frac{1}{n!}\left| \begin{array}{llllll}\mbox{tr}(A) & 1 & 0 & \ldots & \ldots & 0 \\ \mbox{tr}(A^{2}) & \mbox{tr}(A) & 2 & 0 & \ldots & 0 \\ \mbox{tr}(A^{3}) & \mbox{tr}(A^{2}) & \mbox{tr}(A) & 3 & & \vdots \\ \vdots & & & & & n-1 \\ \mbox{tr}(A^{n}) & \mbox{tr}(A^{n-1}) & \mbox{tr}(A^{n-2}) & \ldots & \ldots & \mbox{tr}(A) \end{array}\right|\]
2017 Miklós Schweitzer, 2
Prove that a field $K$ can be ordered if and only if every $A\in M_n(K)$ symmetric matrix can be diagonalized over the algebraic closure of $K$. (In other words, for all $n\in\mathbb{N}$ and all $A\in M_n(K)$, there exists an $S\in GL_n(\overline{K})$ for which $S^{-1}AS$ is diagonal.)
2004 District Olympiad, 4
Let $A=(a_{ij})\in \mathcal{M}_p(\mathbb{C})$ such that $a_{12}=a_{23}=\ldots=a_{p-1,p}=1$ and $a_{ij}=0$ for any other entry.
a)Prove that $A^{p-1}\neq O_p$ and $A^p=O_p$.
b)If $X\in \mathcal{M}_{p}(\mathbb{C})$ and $AX=XA$, prove that there exist $a_1,a_2,\ldots,a_p\in \mathbb{C}$ such that:
\[X=\left( \begin{array}{ccccc} a_1 & a_2 & a_3 & \ldots & a_p \\ 0 & a_1 & a_2 & \ldots & a_{p-1} \\ 0 & 0 & a_1 & \ldots & a_{p-2} \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & a_1 \end{array} \right)\]
c)If there exist $B,C\in \mathcal{M}_p(\mathbb{C})$ such that $(I_p+A)^n=B^n+C^n,\ (\forall)n\in \mathbb{N}^*$, prove that $B=O_p$ or $C=O_p$.
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, 1
At the vertices $A, B, C, D, E, F, G, H$ of a cube, $2001, 2002, 2003, 2004, 2005, 2008, 2007$ and $2006$ stones respectively are placed. It is allowed to move a stone from a vertex to each of its three neighbours, or to move a stone to a vertex from each of its three neighbours. Which of the following arrangements of stones at $A, B, \ldots , H$ can be obtained?
$(\text{a})\quad 2001, 2002, 2003, 2004, 2006, 2007, 2008, 2005;$
$(\text{b})\quad 2002, 2003, 2004, 2001, 2006, 2005, 2008, 2007;$
$(\text{c})\quad 2004, 2002, 2003, 2001, 2005, 2008, 2007, 2006.$