Found problems: 823
2004 Gheorghe Vranceanu, 4
Let be a $ 3\times 3 $ complex matrix such that $ A^3=I $ and for which exist four real numbers $ a,b,c,d $ with $ a,c\neq 1 $ such that $ \det \left( A^2+aA+bI \right) =\det \left( A^2+cA+dI \right) =0. $ Show that $ a+b=c+d. $
[i]C. Merticaru[/i]
2004 USA Team Selection Test, 5
Let $A = (0, 0, 0)$ in 3D space. Define the [i]weight[/i] of a point as the sum of the absolute values of the coordinates. Call a point a [i]primitive lattice point[/i] if all of its coordinates are integers whose gcd is 1. Let square $ABCD$ be an [i]unbalanced primitive integer square[/i] if it has integer side length and also, $B$ and $D$ are primitive lattice points with different weights. Prove that there are infinitely many unbalanced primitive integer squares such that the planes containing the squares are not parallel to each other.
2009 Mediterranean Mathematics Olympiad, 3
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]
2022 Romania National Olympiad, P2
Let $\mathcal{F}$ be the set of pairs of matrices $(A,B)\in\mathcal{M}_2(\mathbb{Z})\times\mathcal{M}_2(\mathbb{Z})$ for which there exists some positive integer $k$ and matrices $C_1,C_2,\ldots, C_k\in\{A,B\}$ such that $C_1C_2\cdots C_k=O_2.$ For each $(A,B)\in\mathcal{F},$ let $k(A,B)$ denote the minimal positive integer $k$ which satisfies the latter property.
[list=a]
[*]Let $(A,B)\in\mathcal{F}$ with $\det(A)=0,\det(B)\neq 0$ and $k(A,B)=p+2$ for some $p\in\mathbb{N}^*.$ Show that $AB^pA=O_2.$
[*]Prove that for any $k\geq 3$ there exists a pair $(A,B)\in\mathcal{F}$ such that $k(A,B)=k.$
[/list][i]Bogdan Blaga[/i]
2025 District Olympiad, P2
Let $n\in\mathbb{Z}$, $n\geq 3$. A matrix $A\in\mathcal{M}_n(\mathbb{C})$ is said to have property $(\mathcal{P})$ if $\det(A+X_{ij})=\det(A+X_{ji})$, for all $i,j\in\{1,2,\dots ,n\}$, where $X_{ij}\in\mathcal{M}_n(\mathbb{C})$ is the matrix with $1$ on position $(i,j)$ and $0$ otherwise.
[list=a]
[*] Show that if $A\in\mathcal{M}_n(\mathbb{C})$ has property $(\mathcal{P})$ and $\det(A)\neq 0$, then $A=A^T$.
[*] Give an example of a matrix $A\in\mathcal{M}_n(\mathbb{C})$ with property $(\mathcal{P})$ such that $A\neq A^T$.
1967 IMO Longlists, 5
Solve the system of equations:
$
\begin{matrix}
x^2 + x - 1 = y \\
y^2 + y - 1 = z \\
z^2 + z - 1 = x.
\end{matrix}
$
2000 District Olympiad (Hunedoara), 2
Calculate the determinant of the $ n\times n $ complex matrix $ \left(a_j^i\right)_{1\le j\le n}^{1\le i\le n} $ defined by
$$ a_j^i=\left\{\begin{matrix} 1+x^2,\quad i=j\\x,\quad |i-j|=1\\0,\quad |i-j|\ge 2\end{matrix}\right. , $$
where $ n $ is a natural number greater than $ 2. $
2011 Romania National Olympiad, 4
[color=darkred]Let $A\, ,\, B\in\mathcal{M}_2(\mathbb{C})$ so that : $A^2+B^2=2AB$ .
[b]a)[/b] Prove that : $AB=BA$ .
[b]b)[/b] Prove that : $\text{tr}\, (A)=\text{tr}\, (B)$ .[/color]
2006 Putnam, B4
Let $Z$ denote the set of points in $\mathbb{R}^{n}$ whose coordinates are $0$ or $1.$ (Thus $Z$ has $2^{n}$ elements, which are the vertices of a unit hypercube in $\mathbb{R}^{n}$.) Given a vector subspace $V$ of $\mathbb{R}^{n},$ let $Z(V)$ denote the number of members of $Z$ that lie in $V.$ Let $k$ be given, $0\le k\le n.$ Find the maximum, over all vector subspaces $V\subseteq\mathbb{R}^{n}$ of dimension $k,$ of the number of points in $V\cap Z.$
2006 Mathematics for Its Sake, 2
Let be a natural number $ n. $ Solve in the set of $ 2\times 2 $ complex matrices the equation
$$ \begin{pmatrix} -2& 2007\\ 0&-2 \end{pmatrix} =X^{3n}-3X^n. $$
[i]Petru Vlad[/i]
2017 Simon Marais Mathematical Competition, A3
For each positive integer $n$, let $M(n)$ be the $n\times n$ matrix whose $(i,j)$ entry is equal to $1$ if $i+1$ is divisible by $j$, and equal to $0$ otherwise. Prove that $M(n)$ is invertible if and only if $n+1$ is square-free. (An integer is [i]square-free[/i] if it is not divisible by a square of an integer larger than $1$.)
2001 Miklós Schweitzer, 8
Let $H$ be a complex Hilbert space. The bounded linear operator $A$ is called [i]positive[/i] if $\langle Ax, x\rangle \geq 0$ for all $x\in H$. Let $\sqrt A$ be the positive square root of $A$, i.e. the uniquely determined positive operator satisfying $(\sqrt{A})^2=A$. On the set of positive operators we introduce the
$$A\circ B=\sqrt A B\sqrt B$$
operation. Prove that for a given pair $A, B$ of positive operators the identity
$$(A\circ B)\circ C=A\circ (B\circ C)$$
holds for all positive operator $C$ if and only if $AB=BA$.
2024 IMC, 7
Let $n$ be a positive integer. Suppose that $A$ and $B$ are invertible $n \times n$ matrices with complex entries such that $A+B=I$ (where $I$ is the identity matrix) and
\[(A^2+B^2)(A^4+B^4)=A^5+B^5.\]
Find all possible values of $\det(AB)$ for the given $n$.
2012 Romania National Olympiad, 3
[color=darkred]Let $A,B\in\mathcal{M}_4(\mathbb{R})$ such that $AB=BA$ and $\det (A^2+AB+B^2)=0$ . Prove that:
\[\det (A+B)+3\det (A-B)=6\det (A)+6\det (B)\ .\][/color]
1950 Miklós Schweitzer, 6
Prove the following identity for determinants:
$ |c_{ik} \plus{} a_i \plus{} b_k \plus{} 1|_{i,k \equal{} 1,...,n} \plus{} |c_{ik}|_{i,k \equal{} 1,...,n} \equal{} |c_{ik} \plus{} a_i \plus{} b_k|_{i,k \equal{} 1,...,n} \plus{} |c_{ik} \plus{} 1|_{i,k \equal{} 1,...,n}$
1997 Federal Competition For Advanced Students, P2, 1
Let $ a$ be a fixed integer. Find all integer solutions $ x,y,z$ of the system:
$ 5x\plus{}(a\plus{}2)y\plus{}(a\plus{}2)z\equal{}a,$
$ (2a\plus{}4)x\plus{}(a^2\plus{}3)y\plus{}(2a\plus{}2)z\equal{}3a\minus{}1,$
$ (2a\plus{}4)x\plus{}(2a\plus{}2)y\plus{}(a^2\plus{}3)z\equal{}a\plus{}1.$
Gheorghe Țițeica 2024, P4
Let $n\geq 2$. Find all matrices $A\in\mathcal{M}_n(\mathbb{C})$ such that $$\text{rank}(A^2)+\text{rank}(B^2)\geq 2\text{rank}(AB),$$ for all $B\in\mathcal{M}_n(\mathbb{C})$.
[i]Cristi Săvescu[/i]
2024 Romanian Master of Mathematics, 3
Given a positive integer $n$, a collection $\mathcal{S}$ of $n-2$ unordered triples of integers in $\{1,2,\ldots,n\}$ is [i]$n$-admissible[/i] if for each $1 \leq k \leq n - 2$ and each choice of $k$ distinct $A_1, A_2, \ldots, A_k \in \mathcal{S}$ we have $$ \left|A_1 \cup A_2 \cup \cdots A_k \right| \geq k+2.$$
Is it true that for all $n > 3$ and for each $n$-admissible collection $\mathcal{S}$, there exist pairwise distinct points $P_1, \ldots , P_n$ in the plane such that the angles of the triangle $P_iP_jP_k$ are all less than $61^{\circ}$ for any triple $\{i, j, k\}$ in $\mathcal{S}$?
[i]Ivan Frolov, Russia[/i]
1991 Arnold's Trivium, 18
Calculate
\[\int\cdots\int \exp\left(-\sum_{1\le i\le j\le n}x_ix_j\right)dx_1\cdots dx_n\]
2000 IMC, 3
Let $A,B\in\mathbb{C}^{n\times n}$ with $\rho(AB - BA) = 1$. Show that $(AB - BA)^2 = 0$.
1985 Traian Lălescu, 1.3
Let be two matrices $ A,B\in M_2\left(\mathbb{R}\right) $ and two natural numbers $ m,n. $ Prove that:
$$ \det\left( (AB)^m-(BA)^m\right)\cdot\det\left( (AB)^n-(BA)^n\right)\ge 0. $$
2006 Cezar Ivănescu, 2
[b]a)[/b] Let $ a,b,c $ be three complex numbers. Prove that the element $ \begin{pmatrix} a & a-b & a-b \\ 0 & b & b-c \\ 0 & 0 & c \end{pmatrix} $ has finite order in the multiplicative group of $ 3\times 3 $ complex matrices if and only if $ a,b,c $ have finite orders in the multiplicative group of complex numbers.
[b]b)[/b] Prove that a $ 3\times 3 $ real matrix $ M $ has positive determinant if there exists a real number $ \lambda\in\left( 0,\sqrt[3]{4} \right) $ such that $ A^3=\lambda A+I. $
[i]Cristinel Mortici[/i]
2022 Romania National Olympiad, P4
Let $A,B\in\mathcal{M}_n(\mathbb{C})$ such that $A^2+B^2=2AB.$ Prove that for any complex number $x$\[\det(A-xI_n)=\det(B-xI_n).\][i]Mihai Opincariu and Vasile Pop[/i]
2006 Moldova MO 11-12, 6
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 $\lim_{n\rightarrow\infty}\frac{x_n}{y_n}$.
2004 Nicolae Coculescu, 4
Let be a matrix $ A\in\mathcal{M}_2(\mathbb{R}) $ having the property that the numbers $ \det (A+X) ,\det (A^2+X^2) ,\det (A^3+X^3) $ are (in this order) in geometric progression, for any matrix $ X\in\mathcal{M}_2(\mathbb{R}) . $
Prove that $ A=0. $
[i]Marius Ghergu[/i]