Found problems: 823
1976 Miklós Schweitzer, 7
Let $ f_1,f_2,\dots,f_n$ be regular functions on a domain of the complex plane, linearly independent over the complex field. Prove that the functions $ f_i\overline{f}_k, \;1 \leq i,k \leq n$, are also linearly independent.
[i]L. Lempert[/i]
1999 IMO Shortlist, 6
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 $x-y$.
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.
2007 Putnam, 5
Let $ k$ be a positive integer. Prove that there exist polynomials $ P_0(n),P_1(n),\dots,P_{k\minus{}1}(n)$ (which may depend on $ k$) such that for any integer $ n,$
\[ \left\lfloor\frac{n}{k}\right\rfloor^k\equal{}P_0(n)\plus{}P_1(n)\left\lfloor\frac{n}{k}\right\rfloor\plus{} \cdots\plus{}P_{k\minus{}1}(n)\left\lfloor\frac{n}{k}\right\rfloor^{k\minus{}1}.\]
($ \lfloor a\rfloor$ means the largest integer $ \le a.$)
2015 District Olympiad, 3
Find all natural numbers $ k\ge 1 $ and $ n\ge 2, $ which have the property that there exist two matrices $ A,B\in M_n\left(\mathbb{Z}\right) $ such that $ A^3=O_n $ and $ A^kB +BA=I_n. $
2012 District Olympiad, 3
Let be a natural number $ n, $ and two matrices $ A,B\in\mathcal{M}_n\left(\mathbb{C}\right) $ with the property that
$$ AB^2=A-B. $$
[b]a)[/b] Show that the matrix $ I_n+B $ is inversable.
[b]b)[/b] Show that $ AB=BA. $
2012 SEEMOUS, Problem 3
a) Prove that if $k$ is an even positive integer and $A$ is a real symmetric $n\times n$ matrix such that $\operatorname{tr}(A^k)^{k+1}=\operatorname{tr}(A^{k+1})^k$, then
$$A^n=\operatorname{tr}(A)A^{n-1}.$$
b) Does the assertion from a) also hold for odd positive integers $k$?
2013 SEEMOUS, Problem 4
Let $A\in M_2(\mathbb Q)$ such that there is $n\in\mathbb N,n\ne0$, with $A^n=-I_2$. Prove that either $A^2=-I_2$ or $A^3=-I_2$.
1940 Putnam, B6
Prove that the determinant of the matrix
$$\begin{pmatrix}
a_{1}^{2}+k & a_1 a_2 & a_1 a_3 &\ldots & a_1 a_n\\
a_2 a_1 & a_{2}^{2}+k & a_2 a_3 &\ldots & a_2 a_n\\
\ldots & \ldots & \ldots & \ldots & \ldots \\
a_n a_1& a_n a_2 & a_n a_3 & \ldots & a_{n}^{2}+k
\end{pmatrix}$$
is divisible by $k^{n-1}$ and find its other factor.
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]
2010 Victor Vâlcovici, 3
Find all positive integers $n \geq 2$ with the following property : there is a matrix $A \in M_{n} (\mathbb{R})$ and a prime number $p \geq 2$ such that $A^{*}$ has exactly $p$ not null elements and $A^{p}=0_{n}$.
2015 IMC, 9
An $n \times n$ complex matrix $A$ is called \emph{t-normal} if
$AA^t = A^t A$ where $A^t$ is the transpose of $A$. For each $n$,
determine the maximum dimension of a linear space of complex $n
\times n$ matrices consisting of t-normal matrices.
Proposed by Shachar Carmeli, Weizmann Institute of Science
1994 IMC, 4
Let $A$ be a $n\times n$ diagonal matrix with characteristic polynomial
$$(x-c_1)^{d_1}(x-c_2)^{d_2}\ldots (x-c_k)^{d_k}$$
where $c_1, c_2, \ldots, c_k$ are distinct (which means that $c_1$ appears $d_1$ times on the diagonal, $c_2$ appears $d_2$ times on the diagonal, etc. and $d_1+d_2+\ldots + d_k=n$).
Let $V$ be the space of all $n\times n$ matrices $B$ such that $AB=BA$. Prove that the dimension of $V$ is
$$d_1^2+d_2^2+\cdots + d_k^2$$
2009 Italy TST, 1
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.
2000 AMC 12/AHSME, 10
The point $ P \equal{} (1,2,3)$ is reflected in the $ xy$-plane, then its image $ Q$ is rotated by $ 180^\circ$ about the $ x$-axis to produce $ R$, and finally, $ R$ is translated by 5 units in the positive-$ y$ direction to produce $ S$. What are the coordinates of $ S$?
$ \textbf{(A)}\ (1,7, \minus{} 3) \qquad \textbf{(B)}\ ( \minus{} 1,7, \minus{} 3) \qquad \textbf{(C)}\ ( \minus{} 1, \minus{} 2,8) \qquad \textbf{(D)}\ ( \minus{} 1,3,3) \qquad \textbf{(E)}\ (1,3,3)$
2006 Iran MO (3rd Round), 3
Suppose $(u,v)$ is an inner product on $\mathbb R^{n}$ and $f: \mathbb R^{n}\longrightarrow\mathbb R^{n}$ is an isometry, that $f(0)=0$.
1) Prove that for each $u,v$ we have $(u,v)=(f(u),f(v)$
2) Prove that $f$ is linear.
2004 Italy TST, 3
Given real numbers $x_i,y_i (i=1,2,\ldots ,n)$, let $A$ be the $n\times n$ matrix given by $a_{ij}=1$ if $x_i\ge y_j$ and $a_{ij}=0$ otherwise. Suppose $B$ is a $n\times n$ matrix whose entries are $0$ and $1$ such that the sum of entries in any row or column of $B$ equals the sum of entries in the corresponding row or column of $A$. Prove that $B=A$.
1965 IMO Shortlist, 2
Consider the sytem of equations
\[ a_{11}x_1+a_{12}x_2+a_{13}x_3 = 0 \]\[a_{21}x_1+a_{22}x_2+a_{23}x_3 =0\]\[a_{31}x_1+a_{32}x_2+a_{33}x_3 = 0 \] with unknowns $x_1, x_2, x_3$. The coefficients satisfy the conditions:
a) $a_{11}, a_{22}, a_{33}$ are positive numbers;
b) the remaining coefficients are negative numbers;
c) in each equation, the sum ofthe coefficients is positive.
Prove that the given system has only the solution $x_1=x_2=x_3=0$.
1991 Arnold's Trivium, 85
Find the lengths of the principal axes of the ellipsoid
\[\sum_{i\le j}x_i x_j=1\]
1981 Spain Mathematical Olympiad, 6
Prove that the transformation product of the symmetry of center $(0, 0)$ with the symmetry of the axis, with the line of equation $x = y + 1$, can be expressed as a product of an axis symmetry the line $e$ by a translation of vector $\overrightarrow{v}$, with $e$ parallel to $\overrightarrow{v}$, .
Determine a line $e$ and a vector $\overrightarrow{v}$, that meet the indicated conditions. have to be unique $e$ and $\overrightarrow{v}$,?
2000 Iran MO (2nd round), 3
Let $M=\{1,2,3,\ldots, 10000\}.$ Prove that there are $16$ subsets of $M$ such that for every $a \in M,$ there exist $8$ of those subsets that intersection of the sets is exactly $\{a\}.$
2012 USA Team Selection Test, 3
Determine, with proof, whether or not there exist integers $a,b,c>2010$ satisfying the equation
\[a^3+2b^3+4c^3=6abc+1.\]
2012 Pre-Preparation Course Examination, 1
Suppose that $W,W_1$ and $W_2$ are subspaces of a vector space $V$ such that $V=W_1\oplus W_2$. Under what conditions we have
$W=(W\cap W_1)\oplus(W\cap W_2)$?
2018 Philippine MO, 3
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.
1950 AMC 12/AHSME, 38
If the expression $ \begin{pmatrix}a & c \\
d & b \end{pmatrix}$ has the value $ ab\minus{}cd$ for all values of $a, b, c$ and $d$, then the equation $ \begin{pmatrix}2x & 1 \\
x & x \end{pmatrix} = 3$:
$\textbf{(A)}\ \text{Is satisfied for only 1 value of }x \qquad\\
\textbf{(B)}\ \text{Is satisified for only 2 values of }x \qquad\\
\textbf{(C)}\ \text{Is satisified for no values of }x \qquad\\
\textbf{(D)}\ \text{Is satisfied for an infinite number of values of }x \qquad\\
\textbf{(E)}\ \text{None of these.}$