Found problems: 638
1989 Irish Math Olympiad, 2
A 3x3 magic square, with magic number $m$, is a $3\times 3$ matrix such that the entries on each row, each column and each diagonal sum to $m$. Show that if the square has positive integer entries, then $m$ is divisible by $3$, and each entry of the square is at most $2n-1$, where $m=3n$. An example of a magic square with $m=6$ is
\[\left( \begin{array}{ccccc}
2 & 1 & 3\\
3 & 2 & 1\\
1 & 3 & 2
\end{array} \right)\]
1996 IMC, 1
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 $\det(A)$.
2013 Vietnam National Olympiad, 1
Solve with full solution:
\[\left\{\begin{matrix}\sqrt{(\sin x)^2+\frac{1}{(\sin x)^2}}+\sqrt{(\cos y)^2+\frac{1}{(\cos y)^2}}=\sqrt\frac{20y}{x+y}
\\\sqrt{(\sin y)^2+\frac{1}{(\sin y)^2}}+\sqrt{(\cos x)^2+\frac{1}{(\cos x)^2}}=\sqrt\frac{20x}{x+y}\end{matrix}\right. \]
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.
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]
1999 Dutch Mathematical Olympiad, 4
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.
2003 Iran MO (3rd Round), 15
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]
1991 Arnold's Trivium, 18
Calculate
\[\int\cdots\int \exp\left(-\sum_{1\le i\le j\le n}x_ix_j\right)dx_1\cdots dx_n\]
2011 Brazil Team Selection Test, 3
2500 chess kings have to be placed on a $100 \times 100$ chessboard so that
[b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex);
[b](ii)[/b] each row and each column contains exactly 25 kings.
Find the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.)
[i]Proposed by Sergei Berlov, Russia[/i]
2005 India IMO Training Camp, 3
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]
2018 District Olympiad, 2
Consider the set
\[M = \left\{
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\in\mathcal{M}_2(\mathbb{C})\ |\ ab = cd
\right\}.\]
a) Give an example of matrix $A\in M$ such that $A^{2017}\in M$ and $A^{2019}\in M$, but $A^{2018}\notin M$.
b) Show that if $A\in M$ and there exists the integer number $k\ge 1$ such that $A^k \in M$, $A^{k + 1}\in M$ si $A^{k + 2} \in M$, then $A^n\in M$, for any integer number $n\ge 1$.
2004 IMO Shortlist, 4
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]
2010 Tournament Of Towns, 5
$33$ horsemen are riding in the same direction along a circular road. Their speeds are constant and pairwise distinct. There is a single point on the road where the horsemen can surpass one another. Can they ride in this fashion for arbitrarily long time ?
2023 Romania National Olympiad, 2
Let $A,B \in M_{n}(\mathbb{R}).$ Show that $rank(A) = rank(B)$ if and only if there exist nonsingular matrices $X,Y,Z \in M_{n}(\mathbb{R})$ such that
\[
AX + YB = AZB.
\]
2016 Romania National Olympiad, 1
Let be a $ 2\times 2 $ real matrix $ A $ that has the property that $ \left| A^d-I_2 \right| =\left| A^d+I_2 \right| , $ for all $ d\in\{ 2014,2016 \} . $
Prove that $ \left| A^n-I_2 \right| =\left| A^n+I_2 \right| , $ for any natural number $ n. $
2014 IMC, 1
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)
2008 Teodor Topan, 1
Solve in $ M_2(\mathbb{C})$ the equation $ X^2\equal{}\left(
\begin{array}{cc}
1 & 2 \\
3 & 6 \end{array}
\right)$
2011 Putnam, A6
Let $G$ be an abelian group with $n$ elements, and let \[\{g_1=e,g_2,\dots,g_k\}\subsetneq G\] be a (not necessarily minimal) set of distinct generators of $G.$ A special die, which randomly selects one of the elements $g_1,g_2,\dots,g_k$ with equal probability, is rolled $m$ times and the selected elements are multiplied to produce an element $g\in G.$
Prove that there exists a real number $b\in(0,1)$ such that \[\lim_{m\to\infty}\frac1{b^{2m}}\sum_{x\in G}\left(\mathrm{Prob}(g=x)-\frac1n\right)^2\] is positive and finite.
2018 Ramnicean Hope, 1
Let be a natural number $ n\ge 2, $ the real numbers $ a_1,a_2,\ldots ,a_n,b_1,b_2,\ldots, b_n, $ and the matrix defined as
$$ A=\left( a_i+b_j \right)_{1\le j\le n}^{1\le i\le n} . $$
[b]a)[/b] Show that $ n=2 $ if $ A $ is invertible.
[b]b)[/b] Prove that the pair of numbers $ a_1,a_2 $ and $ b_1,b_2 $ are both consecutive (not necessarily in this order), if $ A $ is an invertible matrix of integers whose inverse is a matrix of integers.
[i]Costică Ambrinoc[/i]
2001 IMO Shortlist, 8
Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.
2004 China Team Selection Test, 2
Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.
2007 IberoAmerican Olympiad For University Students, 4
Consider an infinite sequence $a_1,a_2,\cdots$ whose terms all belong to $\left\{1,2\right\}$. A positive integer with $n$ digits is said to be [i]good[/i] if its decimal representation has the form $a_ra_{r+1}\cdots a_{r+(n-1)}$, for some positive integer $r$. Suppose that there are at least $2008$ [i]good[/i] numbers with a million digits. Prove that there are at least $2008$ [i]good[/i] numbers with $2007$ digits.
2022 CIIM, 2
Let $v \in \mathbb{R}^2$ a vector of length 1 and $A$ a $2 \times 2$ matrix with real entries such that:
(i) The vectors $A v, A^2 v$ y $A^3 v$ are also of length 1.
(ii) The vector $A^2 v$ isn't equal to $\pm v$ nor to $\pm A v$.
Prove that $A^t A=I_2$.
2013 Singapore Senior Math Olympiad, 4
In the following $6\times 6$ matrix, one can choose any $k\times k$ submatrix, with $1<k\leq6 $ and add $1$ to all its entries. Is it possible to perform the operation a finite number of times so that all the entries in the $6\times 6$ matrix are multiples of $3$?
$ \begin{pmatrix}
2 & 0 & 1 & 0 & 2 & 0 \\
0 & 2 & 0 & 1 & 2 & 0 \\
1 & 0 & 2 & 0 & 2 & 0 \\
0 & 1 & 0 & 2 & 2 & 0 \\
1 & 1 & 1 & 1 & 2 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} $
Note: A $p\times q$ submatrix of a $m\times n$ matrix (with $p\leq m$, $q\leq n$) is a $p\times q$ matrix formed by taking a block of the entries of this size from the original matrix.
2010 Contests, 4
Let $n$ be a positive integer. Find the smallest positive integer $k$ with the property that for any colouring nof the squares of a $2n$ by $k$ chessboard with $n$ colours, there are $2$ columns and $2$ rows such that the $4$ squares in their intersections have the same colour.