Found problems: 638
2006 Iran MO (3rd Round), 1
Suppose that $A\in\mathcal M_{n}(\mathbb R)$ with $\text{Rank}(A)=k$. Prove that $A$ is sum of $k$ matrices $X_{1},\dots,X_{k}$ with $\text{Rank}(X_{i})=1$.
2013 Argentina Cono Sur TST, 4
Show that the number $\begin{matrix} \\ N= \end{matrix} \underbrace{44 \ldots 4}_{n} \underbrace{88 \ldots 8}_{n} - 1\underbrace{33 \ldots3 }_{n-1}2$ is a perfect square for all positive integers $n$.
2016 Korea USCM, 3
Given positive integers $m,n$ and a $m\times n$ matrix $A$ with real entries.
(1) Show that matrices $X = I_m + AA^T$ and $Y = I_n + A^T A$ are invertible. ($I_l$ is the $l\times l$ unit matrix.)
(2) Evaluate the value of $\text{tr}(X^{-1}) - \text{tr}(Y^{-1})$.
1995 Italy TST, 2
Twenty-one rectangles of size $3\times 1$ are placed on an $8\times 8$ chessboard, leaving only one free unit square. What position can the free square lie at?
2002 Putnam, 6
Let $p$ be a prime number. Prove that the determinant of the matrix \[ \begin{bmatrix}x & y & z\\ x^p & y^p & z^p \\ x^{p^2} & y^{p^2} & z^{p^2} \end{bmatrix} \] is congruent modulo $p$ to a product of polynomials of the form $ax+by+cz$, where $a$, $b$, and $c$ are integers. (We say two integer polynomials are congruent modulo $p$ if corresponding coefficients are congruent modulo $p$.)
1958 AMC 12/AHSME, 40
Given $ a_0 \equal{} 1$, $ a_1 \equal{} 3$, and the general relation $ a_n^2 \minus{} a_{n \minus{} 1}a_{n \plus{} 1} \equal{} (\minus{}1)^n$ for $ n \ge 1$. Then $ a_3$ equals:
$ \textbf{(A)}\ \frac{13}{27}\qquad
\textbf{(B)}\ 33\qquad
\textbf{(C)}\ 21\qquad
\textbf{(D)}\ 10\qquad
\textbf{(E)}\ \minus{}17$
1985 IMO Longlists, 80
Let $E = \{1, 2, \dots , 16\}$ and let $M$ be the collection of all $4 \times 4$ matrices whose entries are distinct members of $E$. If a matrix $A = (a_{ij} )_{4\times4}$ is chosen randomly from $M$, compute the probability $p(k)$ of $\max_i \min_j a_{ij} = k$ for $k \in E$. Furthermore, determine $l \in E$ such that $p(l) = \max \{p(k) | k \in E \}.$
1991 Putnam, A2
$M$ and $N$ are real unequal $n\times n$ matrices satisfying $M^3=N^3$ and $M^2N=N^2M$. Can we choose $M$ and $N$ so that $M^2+N^2$ is invertible?
2006 Bulgaria Team Selection Test, 1
[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]
1990 Greece National Olympiad, 1
Let $A$ be a $2\,x\,2$ matrix with real numbers. Prove that if $A^3=\mathbb{O}$ then $A^2=\mathbb{O}$.
2003 VJIMC, Problem 2
Let $A=(a_{ij})$ be an $m\times n$ real matrix with at least one non-zero element. For each $i\in\{1,\ldots,m\}$, let $R_i=\sum_{j=1}^na_{ij}$ be the sum of the $i$-th row of the matrix $A$, and for each $j\in\{1,\ldots,n\}$, let $C_j =\sum_{i=1}^ma_{ij}$ be the sum of the $j$-th column of the matrix $A$. Prove that there exist indices $k\in\{1,\ldots,m\}$ and $l\in\{1,\ldots,n\}$ such that
$$a_{kl}>0,\qquad R_k\ge0,\qquad C_l\ge0,$$or
$$a_{kl}<0,\qquad R_k\le0,\qquad C_l\le0.$$
1995 IMC, 1
Let $X$ be a invertible matrix with columns $X_{1},X_{2}...,X_{n}$. Let $Y$ be a matrix with columns $X_{2},X_{3},...,X_{n},0$. Show that the matrices $A=YX^{-1}$ and $B=X^{-1}Y$ have rank $n-1$ and have only $0$´s for eigenvalues.
2018 Korea USCM, 5
A real symmetric $2018\times 2018$ matrix $A=(a_{ij})$ satisfies $|a_{ij}-2018|\leq 1$ for every $1\leq i,j\leq 2018$. Denote the largest eigenvalue of $A$ by $\lambda(A)$. Find maximum and minumum value of $\lambda(A)$.