Determine all pairs $(a, b)$ of real numbers for which there exists a unique symmetric $2\times 2$ matrix $M$ with real entries satisfying $\mathrm{trace}(M)=a$ and $\mathrm{det}(M)=b$. (Proposed by Stephan Wagner, Stellenbosch University)

Consider all parabolas of the form $ y\equal{}x^2\plus{}2px\plus{}q$ for $ p,q \in \mathbb{R}$ which intersect the coordinate axes in three distinct points. For such $ p,q$, denote by $ C_{p,q}$ the circle through these three intersection points. Prove that all circles $ C_{p,q}$ have a point in common.

Let $p$ be a prime number. Prove that from a $p^2\times p^2$ array of squares, we can select $p^3$ of the squares such that the centers of any four of the selected squares are not the vertices of a rectangle with sides parallel to the edges of the array.

Let $n$ be a positive integer. For $i$ and $j$ in $\{1,2, \ldots, n\}$, let $s(i, j)$ be the number of pairs $(a, b)$ of nonnegative integers satisfying $a i+b j=n$. Let $S$ be the $n$-by-n matrix whose $(i, j)$-entry is $s(i, j)$. For example, when $n=5$, we have $S=\left[\begin{array}{lllll}6 & 3 & 2 & 2 & 2 \\ 3 & 0 & 1 & 0 & 1 \\ 2 & 1 & 0 & 0 & 1 \\ 2 & 0 & 0 & 0 & 1 \\ 2 & 1 & 1 & 1 & 2\end{array}\right]$. Compute the determinant of $S$.

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]

Let $A$ be a $5\times10$ matrix with real entries, and let $A^{\text T}$ denote its transpose. Suppose every $5\times1$ matrix with real entries can be written in the form $A\mathbf u$ where $\mathbf u$ is a $10\times1$ matrix with real entries. Prove that every $5\times1$ matrix with real entries can be written in the form $AA^{\text T}\mathbf v$ where $\mathbf v$ is a $5\times1$ matrix with real entries.

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)$.

A matrix $A=(a_{ij})$ is called [i]nice[/i], if it has the following properties: (i) the set of all entries of $A$ is $\{1,2,\dots,2t\}$ for some integer $t$; (ii) the entries are non-decreasing in every row and in every column: $a_{i,j} \le a_{i,j+1}$ and $a_{i,j} \le a_{i+1,j}$; (iii) equal entries can appear only in the same row or the same column: if $a_{i,j}=a_{k,\ell}$, then either $i=k$ or $j=\ell$; (iv) for each $s=1,2,\dots,2t-1$, there exist $i \ne k$ and $j \ne \ell$ such that $a_{i,j}=s$ and $a_{k,\ell}=s+1$. Prove that for any positive integers $m$ and $n$, the number of nice $m \times n$ matrixes is even. For example, the only two nice $2 \times 3$ matrices are $\begin{pmatrix} 1 & 1 & 1\\2 & 2 & 2 \end{pmatrix}$ and $\begin{pmatrix} 1 & 1 & 3\\2 & 4 & 4 \end{pmatrix}$.

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 $V\cap Z.$

Let $n \geq 2$ and $A, B \in \mathcal{M}_n(\mathbb{C})$ such that there exists an idempotent matrix $C \in \mathcal{M}_n(\mathbb{C})$ for which $C^*=AB-BA.$ Prove that $(AB-BA)^2=0.$ Note: $X^*$ is the [url = https://en.wikipedia.org/wiki/Adjugate_matrix]adjugate[/url] matrix of $X$ (not the conjugate transpose)

Let be two $ 3\times 3 $ real matrices that have the property that $$ AX=\begin{pmatrix}0\\0\\0\end{pmatrix}\implies BX=\begin{pmatrix}0\\0\\0\end{pmatrix} , $$ for any three-dimensional vectors $ X. $ Prove that there exists a $ 3\times 3 $ real matrix $ C $ such that $ B=CA. $

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]

Let be two matrices $ A,B\in M_2\left(\mathbb{R}\right) $ that satisfy the equality $ \left( A-B\right)^2 =O_2. $ [b]a)[/b] Show that $ \det\left( A^2-B^2\right) =\left( \det A -\det B\right)^2. $ [b]b)[/b] Demonstrate that $ \det\left( AB-BA\right) =0\iff \det A=\det B. $

Suppose that every integer has been given one of the colours red, blue, green or yellow. Let $x$ and $y$ be odd integers so that $|x| \neq |y|$. Show that there are two integers of the same colour whose difference has one of the following values: $x,y,x+y$ or $x-y$.

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 $\det(A)$.

Given the vector spaces $V,W$ with coefficients over a field $K$ and function $ \phi :V\to W$ satisfying the relation : $$\varphi(\lambda x+y)= \lambda \varphi(x)+\phi (y)$$ for all $x,y \in V, \lambda \in K$. Such a function is called linear. Let $L\varphi=\{x\in V/\varphi(x)=0\}$ , and$M=\varphi(V)$ , prove that : (i) $L\varphi$ is subspace of $V$ and $M$ is subspace of $W$ (ii) $L\varphi={O}$ iff $\varphi$ is $1-1$ (iii) Dimension of $V$ equals to dimension of $L\varphi$ plus dimension of $M$ (iv) If $\theta : \mathbb{R}^3\to\mathbb{R}^3$ with $\theta(x,y,z)=(2x-z,x-y,x-3y+z)$, prove that $\theta$ is linear function . Find $L\theta=\{x\in {R}^3/\theta(x)=0\}$ and dimension of $M=\theta({R}^3)$.

Let $A,B\in \mathcal{M}_n(\mathbb{C})$ such that $AB=BA$ and $\det B\neq 0$. a) If $|\det(A+zB)|=1$ for any $z\in \mathbb{C}$ such that $|z|=1$, then $A^n=O_n$. b) Is the question from a) still true if $AB\neq BA$ ?

Solve with full solution: \[\left\{\begin{matrix}\sqrt{(\sin x)^2+\frac{1}{(\sin x)^2}}+\sqrt{(\cos y)^2+\frac{1}{(\cos y)^2}}=\sqrt\frac{20y}{x+y} \\\sqrt{(\sin y)^2+\frac{1}{(\sin y)^2}}+\sqrt{(\cos x)^2+\frac{1}{(\cos x)^2}}=\sqrt\frac{20x}{x+y}\end{matrix}\right. \]

Let $n,k$ be positive integers such that $n\ge k$. $n$ lamps are placed on a circle, which are all off. In any step we can change the state of $k$ consecutive lamps. In the following three cases, how many states of lamps are there in all $2^n$ possible states that can be obtained from the initial state by a certain series of operations? i)$k$ is a prime number greater than $2$; ii) $k$ is odd; iii) $k$ is even.

Let be two $ 2\times 2 $ real matrices $ A,B, $ such that $ AB=\begin{pmatrix} 1&1\\1&2 \end{pmatrix} . $ Calculate $ \left((BA)^{-1} +BA\right)^{2019 } . $ [i]Dan Nedeianu[/i]

[b]Problem 1. [/b]In the cells of square table are written the numbers $1$, $0$ or $-1$ so that in every line there is exactly one $1$, amd exactly one $-1$. Each turn we change the places of two columns or two rows. Is it possible, from any such table, after finite number of turns to obtain its opposite table (two tables are opposite if the sum of the numbers written in any two corresponding squares is zero)? [i] Emil Kolev[/i]

Consider a matrix of size $8 \times 8$, containing positive integers only. One may repeatedly transform the entries of the matrix according to the following rules: -Multiply all entries in some row by 2. -Subtract 1 from all entries in some column. Prove that one can transform the given matrix into the zero matrix.

Assume $m\times n$ matrix which is filled with just 0, 1 and any two row differ in at least $n/2$ members, show that $m \leq 2n$. ( for example the diffrence of this two row is only in one index 110 100) [i]Edited by Myth[/i]

Calculate \[\int\cdots\int \exp\left(-\sum_{1\le i\le j\le n}x_ix_j\right)dx_1\cdots dx_n\]

Let be two distinct $ 2\times 2 $ real matrices having the property that there exists a natural number such that these matrices raised to this number are equal, and these matrices raised to the successor of this number are also equal. Prove that these matrices raised to any power greater than $ 2 $ are equal.