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]

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.

For $ t \equal{} 1, 2, 3, 4$, define $ \displaystyle S_t \equal{} \sum_{i \equal{} 1}^{350}a_i^t$, where $ a_i \in \{1,2,3,4\}$. If $ S_1 \equal{} 513$ and $ S_4 \equal{} 4745$, find the minimum possible value for $ S_2$.

Let $n$ be a nonzero natural number and $f:\mathbb R\to\mathbb R\setminus\{0\}$ be a function such that $f(2014)=1-f(2013)$. Let $x_1,x_2,x_3,\ldots,x_n$ be real numbers not equal to each other. If $$\begin{vmatrix}1+f(x_1)&f(x_2)&f(x_3)&\cdots&f(x_n)\\f(x_1)&1+f(x_2)&f(x_3)&\cdots&f(x_n)\\f(x_1)&f(x_2)&1+f(x_3)&\cdots&f(x_n)\\\vdots&\vdots&\vdots&\ddots&\vdots\\f(x_1)&f(x_2)&f(x_3)&\cdots&1+f(x_n)\end{vmatrix}=0,$$prove that $f$ is not continuous.

Let $f_n\ (n=1,\ 2,\ \cdots)$ be a linear transformation expressed by a matrix $\left( \begin{array}{cc} 1-n & 1 \\ -n(n+1) & n+2 \end{array} \right)$ on the $xy$ plane. Answer the following questions: (1) Prove that there exists 2 lines passing through the origin $O(0,\ 0)$ such that all points of the lines are mapped to the same lines, then find the equation of the lines. (2) Find the area $S_n$ of the figure enclosed by the lines obtained in (1) and the curve $y=x^2$. (3) Find $\sum_{n=1}^{\infty} \frac{1}{S_n-\frac 16}.$ [i]2011 Tokyo Institute of Technlogy entrance exam, Problem 1[/i]

Let be a $ 2\times 2 $ real matrix $ A $ whose primary diagonal has positive elements and whose secondary diagonal has negative elements. If $ \det A>0, $ show that [b]a)[/b] for any $ 2\times 2 $ matrix $ X $ of positive real numbers there exists a $ 2\times 2 $ matrix of positive real numbers such that $ AY=X. $ [b]b)[/b] there is a $ 2\times 2 $ matrix $ Z $ of positive real numbers having the property that all elements of $ AZ $ are positive. [i]Vasile Pop[/i]

Let $ U$ be an $ n \times n$ orthogonal matrix. Prove that for any $ n \times n$ matrix $ A$, the matrices \[ A_m=\frac{1}{m+1} \sum_{j=0}^m U^{-j}AU^j\] converge entrywise as $ m \rightarrow \infty.$ [i]L. Kovacs[/i]

Let $ X: \equal{} \{x_1,x_2,\ldots,x_{29}\}$ be a set of $ 29$ boys: they play with each other in a tournament of Pro Evolution Soccer 2009, in respect of the following rules: [list]i) every boy play one and only one time against each other boy (so we can assume that every match has the form $ (x_i \text{ Vs } x_j)$ for some $ i \neq j$); ii) if the match $ (x_i \text{ Vs } x_j)$, with $ i \neq j$, ends with the win of the boy $ x_i$, then $ x_i$ gains $ 1$ point, and $ x_j$ doesn’t gain any point; iii) if the match $ (x_i \text{ Vs } x_j)$, with $ i \neq j$, ends with the parity of the two boys, then $ \frac {1}{2}$ point is assigned to both boys. [/list] (We assume for simplicity that in the imaginary match $ (x_i \text{ Vs } x_i)$ the boy $ x_i$ doesn’t gain any point). Show that for some positive integer $ k \le 29$ there exist a set of boys $ \{x_{t_1},x_{t_2},\ldots,x_{t_k}\} \subseteq X$ such that, for all choice of the positive integer $ i \le 29$, the boy $ x_i$ gains always a integer number of points in the total of the matches $ \{(x_i \text{ Vs } x_{t_1}),(x_i \text{ Vs } x_{t_2}),\ldots, (x_i \text{ Vs } x_{t_k})\}$. [i](Paolo Leonetti)[/i]

Let $n$ be a positive integer. Consider a triangular array of nonnegative integers as follows: \[ \begin{array}{rccccccccc} \text{Row } 1: &&&&& a_{0,1} &&&& \smallskip\\ \text{Row } 2: &&&& a_{0,2} && a_{1,2} &&& \smallskip\\ &&& \vdots && \vdots && \vdots && \smallskip\\ \text{Row } n-1: && a_{0,n-1} && a_{1,n-1} && \cdots && a_{n-2,n-1} & \smallskip\\ \text{Row } n: & a_{0,n} && a_{1,n} && a_{2,n} && \cdots && a_{n-1,n} \end{array} \] Call such a triangular array [i]stable[/i] if for every $0 \le i < j < k \le n$ we have \[ a_{i,j} + a_{j,k} \le a_{i,k} \le a_{i,j} + a_{j,k} + 1. \] For $s_1, \ldots s_n$ any nondecreasing sequence of nonnegative integers, prove that there exists a unique stable triangular array such that the sum of all of the entries in row $k$ is equal to $s_k$.

Let $n\ge1$. Assume that $A$ is a real $n\times n$ matrix which satisfies the equality $$A^7+A^5+A^3+A-I=0.$$ Show that $\det(A)>0$.

Answer wither (i) or (ii): (i) Let $V, V_1 , V_2$ and $V_3$ denote four vertices of a cube such that $V_1 , V_2 , V_3 $ are adjacent to $V.$ Project the cube orthogonally on a plane of which the points are marked with complex numbers. Let the projection of $V$ fall in the origin and the projections of $V_1 , V_2 , V_3 $ in points marked with the complex numbers $z_1 , z_2 , z_3$, respectively. Show that $z_{1}^{2} +z_{2}^{2} +z_{3}^{2}=0.$ (ii) Let $(a_{ij})$ be a matrix such that $$|a_{ii}| > |a_{i1}| + |a_{i2}|+\ldots +|a_{i i-1}|+ |a_{i i+1}| +\ldots +|a_{in}|$$ for all $i.$ Show that the determinant is not equal to $0.$

Let $ A_1,A_2,\dots,A_n$ be idempotent matrices with real entries. Prove that: \[ \mbox{N}(A_1)\plus{}\mbox{N}(A_2)\plus{}\dots\plus{}\mbox{N}(A_n)\geq \mbox{rank}(I\minus{}A_1A_2\dots A_n)\] $ \mbox{N}(A)$ is $ \mbox{dim}(\mbox{ker(A)})$

Alice and Bob play the following game: Initially, there is a $2016\times2016$ "empty" matrix. Taking turns, with Alice playing first, each player chooses a real number and fill it into an empty entry. If the determinant of the last matrix is non-zero, then Alice wins. Otherwise, Bob wins. Who has the winning strategy?

In the class, there are $ 15$ boys and $ 15$ girls. On March $ 8$, some boys made phone calls to some girls to congratulate them on the holiday ( each boy made no more than one call to each girl). It appears that there is a unique way to split the class in $ 15$ pairs (each consisting of a boy and a girl) such that in every pair the boy has phoned the girl. Find the maximal possible number of calls.