Solve the following system of equations in $\mathbb{R}$ where all square roots are non-negative: $ \begin{matrix} a - \sqrt{1-b^2} + \sqrt{1-c^2} = d \\ b - \sqrt{1-c^2} + \sqrt{1-d^2} = a \\ c - \sqrt{1-d^2} + \sqrt{1-a^2} = b \\ d - \sqrt{1-a^2} + \sqrt{1-b^2} = c \\ \end{matrix} $

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.

We say that the rank of a group $ G$ is at most $ r$ if every subgroup of $ G$ can be generated by at most $ r$ elements. Prove that here exists an integer $ s$ such that for every finite group $ G$ of rank $ 2$ the commutator series of $ G$ has length less than $ s$. [i]J. Erdos[/i]

Let $n$ be a positive integer. An $n \times n$ matrix (a rectangular array of numbers with $n$ rows and $n$ columns) is said to be a platinum matrix if: [list=i] [*] the $n^2$ entries are integers from $1$ to $n$; [*] each row, each column, and the main diagonal (from the upper left corner to the lower right corner) contains each integer from $1$ to $n$ exactly once; and [*] there exists a collection of $n$ entries containing each of the numbers from $1$ to $n$, such that no two entries lie on the same row or column, and none of which lie on the main diagonal of the matrix. [/list] Determine all values of $n$ for which there exists an $n \times n$ platinum matrix.

Let be three complex numbers $ \alpha ,\beta ,\gamma $ such that $$ \begin{vmatrix} \left( \alpha -\beta \right)^2 & \left( \alpha -\beta \right)\left( \beta -\gamma \right) & \left( \beta -\gamma \right)^2 \\ \left( \beta -\gamma \right)^2 & \left( \beta -\gamma \right)\left( \gamma -\alpha \right) & \left( \gamma -\alpha \right)^2 \\ \left( \gamma -\alpha \right)^2 & \left( \gamma -\alpha \right)\left( \alpha -\beta \right) & \left( \alpha -\beta \right)^2\end{vmatrix} =0. $$ Prove that $ \alpha ,\beta ,\gamma $ are all equal, or their affixes represent a non-degenerate equilateral triangle. [i]Gheorghe Necșuleu[/i] and [i]Ion Necșuleu[/i]

$A,B \in M_{2\times 2}(C)$ Prove that: $Tr(AAABBABAABBB)=tr(BBBAABABBAAA)$

$ 15 $ minors of order $ 3 $ of a $ 4\times 4 $ real matrix whose determinant is a nonzero rational number, are rational. Prove that this matrix is rational.

Find the critical values and critical points of the mapping $z\mapsto z^2+2\overline{z}$ (sketch the answer).

Let $A$ be an $n\times n$ complex matrix such that $A \ne \lambda I_{n}$ for all $\lambda \in \mathbb{C}$. Prove that $A$ is similar to a matrix having at most one non-zero entry on the maindiagonal.

Let $ n$ and $ k$ be positive integers. Say that a permutation $ \sigma$ of $ \{1,2,\dots n\}$ is $ k$-[i]limited[/i] if $ |\sigma(i)\minus{}i|\le k$ for all $ i.$ Prove that the number of $ k$-limited permutations of $ \{1,2,\dots n\}$ is odd if and only if $ n\equiv 0$ or $ 1\pmod{2k\plus{}1}.$

Let $A, B \in \mathcal{M}_n(\mathbb{C})$ be such that $AB^2A = AB$. Prove that: a) $(AB)^2 = AB.$ b) $(AB - BA)^3 = O_n.$

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 $n$.

$f: \mathbb R^{n}\longrightarrow\mathbb R^{n}$ is a bijective map, that Image of every $n-1$-dimensional affine space is a $n-1$-dimensional affine space. 1) Prove that Image of every line is a line. 2) Prove that $f$ is an affine map. (i.e. $f=goh$ that $g$ is a translation and $h$ is a linear map.)

Prove that the elements of any natural power of a $ 2\times 2 $ special linear integer matrix are pairwise coprime, with the possible exception of the pairs that form the diagonals. [i]Vasile Pop[/i]

Let be a natural number $ n $ and a matrix $ A\in\mathcal{M}_n(\mathbb{R}) $ having the property that sum of the squares of all its elements is strictly less than $ 1. $ Prove that the matrices $ I\pm A $ are invertible.

a) Show that $\forall n \in \mathbb{N}_0, \exists A \in \mathbb{R}^{n\times n}: A^3=A+I$. b) Show that $\det(A)>0, \forall A$ fulfilling the above condition.

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]

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.$

Let $A,B\in \mathcal{M}_n(\mathbb{R})$ such that $B^2=I_n$ and $A^2=AB+I_n$. Prove that: \[\det A\le \left(\frac{1+\sqrt{5}}{2}\right)^n\]

suppose that $p$ is a prime number. find that smallest $n$ such that there exists a non-abelian group $G$ with $|G|=p^n$. SL is an acronym for Special Lesson. this year our special lesson was Groups and Symmetries. the exam time was 5 hours.

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 $a+b+c=2014$.

Let $n$ be a positive integer. Given a sequence $\varepsilon_1$, $\dots$, $\varepsilon_{n - 1}$ with $\varepsilon_i = 0$ or $\varepsilon_i = 1$ for each $i = 1$, $\dots$, $n - 1$, the sequences $a_0$, $\dots$, $a_n$ and $b_0$, $\dots$, $b_n$ are constructed by the following rules: \[a_0 = b_0 = 1, \quad a_1 = b_1 = 7,\] \[\begin{array}{lll} a_{i+1} = \begin{cases} 2a_{i-1} + 3a_i, \\ 3a_{i-1} + a_i, \end{cases} & \begin{array}{l} \text{if } \varepsilon_i = 0, \\ \text{if } \varepsilon_i = 1, \end{array} & \text{for each } i = 1, \dots, n - 1, \\[15pt] b_{i+1}= \begin{cases} 2b_{i-1} + 3b_i, \\ 3b_{i-1} + b_i, \end{cases} & \begin{array}{l} \text{if } \varepsilon_{n-i} = 0, \\ \text{if } \varepsilon_{n-i} = 1, \end{array} & \text{for each } i = 1, \dots, n - 1. \end{array}\] Prove that $a_n = b_n$. [i]Proposed by Ilya Bogdanov, Russia[/i]

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 $AB=O_n$.

In each square of a chessboard with $a$ rows and $b$ columns, a $0$ or $1$ is written satisfying the following conditions. [list][*]If a row and a column intersect in a square with a $0$, then that row and column have the same number of $0$s. [*]If a row and a column intersect in a square with a $1$, then that row and column have the same number of $1$s.[/list] Find all pairs $(a,b)$ for which this is possible.

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 $n\ge 3$.