Found problems: 638
2006 District Olympiad, 2
Let $n,p \geq 2$ be two integers and $A$ an $n\times n$ matrix with real elements such that $A^{p+1} = A$.
a) Prove that $\textrm{rank} \left( A \right) + \textrm{rank} \left( I_n - A^p \right) = n$.
b) Prove that if $p$ is prime then \[ \textrm{rank} \left( I_n - A \right) = \textrm{rank} \left( I_n - A^2 \right) = \ldots = \textrm{rank} \left( I_n - A^{p-1} \right) . \]
2010 IberoAmerican Olympiad For University Students, 5
Let $A,B$ be matrices of dimension $2010\times2010$ which commute and have real entries, such that $A^{2010}=B^{2010}=I$, where $I$ is the identity matrix. Prove that if $\operatorname{tr}(AB)=2010$, then $\operatorname{tr}(A)=\operatorname{tr}(B)$.
2002 Iran Team Selection Test, 10
Suppose from $(m+2)\times(n+2)$ rectangle we cut $4$, $1\times1$ corners. Now on first and last row first and last columns we write $2(m+n)$ real numbers. Prove we can fill the interior $m\times n$ rectangle with real numbers that every number is average of it's $4$ neighbors.
2011 USAMO, 6
Let $A$ be a set with $|A|=225$, meaning that $A$ has 225 elements. Suppose further that there are eleven subsets $A_1, \ldots, A_{11}$ of $A$ such that $|A_i|=45$ for $1\leq i\leq11$ and $|A_i\cap A_j|=9$ for $1\leq i<j\leq11$. Prove that $|A_1\cup A_2\cup\ldots\cup A_{11}|\geq 165$, and give an example for which equality holds.
2007 All-Russian Olympiad Regional Round, 8.8
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.
2002 Iran Team Selection Test, 10
Suppose from $(m+2)\times(n+2)$ rectangle we cut $4$, $1\times1$ corners. Now on first and last row first and last columns we write $2(m+n)$ real numbers. Prove we can fill the interior $m\times n$ rectangle with real numbers that every number is average of it's $4$ neighbors.
1987 AMC 12/AHSME, 25
$ABC$ is a triangle: $A=(0,0)$, $B=(36,15)$ and both the coordinates of $C$ are integers. What is the minimum area $\triangle ABC$ can have?
$ \textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \frac{3}{2} \qquad\textbf{(D)}\ \frac{13}{2} \qquad\textbf{(E)}\ \text{there is no minimum} $
2002 India IMO Training Camp, 6
Determine the number of $n$-tuples of integers $(x_1,x_2,\cdots ,x_n)$ such that $|x_i| \le 10$ for each $1\le i \le n$ and $|x_i-x_j| \le 10$ for $1 \le i,j \le n$.
2009 Finnish National High School Mathematics Competition, 1
In a plane, the point $(x,y)$ has temperature $x^2+y^2-6x+4y$. Determine the coldest point of the plane and its temperature.
2019 VJIMC, 3
For an invertible $n\times n$ matrix $M$ with integer entries we define a sequence $\mathcal{S}_M=\{M_i\}_{i=0}^{\infty}$ by the recurrence $M_0=M$ ,$M_{i+1}=(M_i^T)^{-1}M_i$ for $i\geq 0$.
Find the smallest integer $n\geq 2 $ for wich there exists a normal $n\times n$ matrix with integer entries such that its sequence $\mathcal{S}_M$ is not constant and has period $P=7$ i.e $M_{i+7}=M_i$.
($M^T$ means the transpose of a matrix $M$ . A square matrix is called normal if $M^T M=M M^T$ holds).
[i]Proposed by Martin Niepel (Comenius University, Bratislava)..[/i]
1957 Putnam, B1
Consider the determinant of the matrix $(a_{ij})_{ij}$ with $1\leq i,j \leq 100$ and $a_{ij}=ij.$ Prove that if the absolute value of each of the $100!$ terms in the expansion of this determinant is divided by $101,$ then the remainder is always $1.$
1976 IMO, 2
We consider the following system
with $q=2p$:
\[\begin{matrix} a_{11}x_{1}+\ldots+a_{1q}x_{q}=0,\\ a_{21}x_{1}+\ldots+a_{2q}x_{q}=0,\\ \ldots ,\\ a_{p1}x_{1}+\ldots+a_{pq}x_{q}=0,\\ \end{matrix}\]
in which every coefficient is an element from the set $\{-1,0,1\}$$.$ Prove that there exists a solution $x_{1}, \ldots,x_{q}$ for the system with the properties:
[b]a.)[/b] all $x_{j}, j=1,\ldots,q$ are integers$;$
[b]b.)[/b] there exists at least one j for which $x_{j} \neq 0;$
[b]c.)[/b] $|x_{j}| \leq q$ for any $j=1, \ldots ,q.$
2020 SEEMOUS, Problem 3
Let $n$ be a positive integer, $k\in \mathbb{C}$ and $A\in \mathcal{M}_n(\mathbb{C})$ such that $\text{Tr } A\neq 0$ and $$\text{rank } A +\text{rank } ((\text{Tr } A) \cdot I_n - kA) =n.$$
Find $\text{rank } A$.
1987 Romania Team Selection Test, 1
Let $a,b,c$ be distinct real numbers such that $a+b+c>0$. Let $M$ be the set of $3\times 3$ matrices with the property that each line and each column contain all given numbers $a,b,c$. Find $\{\max \{ \det A \mid A \in M \}$ and the number of matrices which realise the maximum value.
[i]Mircea Becheanu[/i]
2011 N.N. Mihăileanu Individual, 2
Let be a natural number $ k, $ and a matrix $ M\in\mathcal{M}_k(\mathbb{R}) $ having the property that
$$ \det\left( I-\frac{1}{n^2}\cdot A^2 \right) +1\ge\det \left( I -\frac{1}{n}\cdot A \right) +\det \left( I +\frac{1}{n}\cdot A \right) , $$
for all natural numbers $ n. $ Prove that the trace of $ A $ is $ 0. $
[i]Nelu Chichirim[/i]
1997 Brazil Team Selection Test, Problem 4
Consider an $N\times N$ matrix, where $N$ is an odd positive integer, such that all its entries are $-1,0$ or $1$. Consider the sum of the numbers in every line and every column. Prove that at least two of the $2N$ sums are equal.
2009 Math Prize For Girls Problems, 17
Let $ a$, $ b$, $ c$, $ x$, $ y$, and $ z$ be real numbers that satisfy the three equations
\begin{align*}
13x + by + cz &= 0 \\
ax + 23y + cz &= 0 \\
ax + by + 42z &= 0.
\end{align*}Suppose that $ a \ne 13$ and $ x \ne 0$. What is the value of
\[ \frac{13}{a - 13} + \frac{23}{b - 23} + \frac{42}{c - 42} \, ?\]
1987 Greece National Olympiad, 2
Let $A=(\alpha_{ij})$ be a $m\,x\,n$ matric and $B=(\beta_{kl})$ be a $n\,x\, m$ matric with $m>n$ . Prove that $D(A\cdot B)=0$.
2001 All-Russian Olympiad, 2
In a magic square $n \times n$ composed from the numbers $1,2,\cdots,n^2$, the centers of any two squares are joined by a vector going from the smaller number to the bigger one. Prove that the sum of all these vectors is zero. (A magic square is a square matrix such that the sums of entries in all its rows and columns are equal.)
2017 Korea USCM, 6
Given a positive integer $n$ and a real valued $n\times n$ matrix $A$. $J$ is $n\times n$ matrix with every entry $1$. Suppose $A$ satisfies the following relations.
$$A+A^T = \frac{1}{n} J, \quad AJ = \frac{1}{2} J$$
Show that $A^m-I$ is an invertible matrix for all positive odd integer $m$.
2021 Brazil Undergrad MO, Problem 5
Find all triplets $(\lambda_1,\lambda_2,\lambda_3) \in \mathbb{R}^3$ such that there exists a matrix $A_{3 \times 3}$ with all entries being non-negative reals whose eigenvalues are $\lambda_1,\lambda_2,\lambda_3$.
2012 IMO Shortlist, C3
In a $999 \times 999$ square table some cells are white and the remaining ones are red. Let $T$ be the number of triples $(C_1,C_2,C_3)$ of cells, the first two in the same row and the last two in the same column, with $C_1,C_3$ white and $C_2$ red. Find the maximum value $T$ can attain.
[i]Proposed by Merlijn Staps, The Netherlands[/i]
2023 Brazil Undergrad MO, 4
Let $M_2(\mathbb{Z})$ be the set of $2 \times 2$ matrices with integer entries. Let $A \in M_2(\mathbb{Z})$ such that $$A^2+5I=0,$$ where $I \in M_2(\mathbb{Z})$ and $0 \in M_2(\mathbb{Z})$ denote the identity and null matrices, respectively. Prove that there exists an invertible matrix $C \in M_2(\mathbb{Z})$ with $C^{-1} \in M_2(\mathbb{Z})$ such that $$CAC^{-1} = \begin{pmatrix} 1 & 2\\ -3 & -1 \end{pmatrix} \text{ ou } CAC^{-1} = \begin{pmatrix} 0 & 1\\ -5 & 0 \end{pmatrix}.$$
2011 Tournament of Towns, 7
In every cell of a square table is a number. The sum of the largest two numbers in each row
is $a$ and the sum of the largest two numbers in each column is b. Prove that $a = b$.
2016 VJIMC, 3
For $n \geq 3$ find the eigenvalues (with their multiplicities) of the $n \times n$ matrix
$$\begin{bmatrix}
1 & 0 & 1 & 0 & 0 & 0 & \dots & \dots & 0 & 0\\
0 & 2 & 0 & 1 & 0 & 0 & \dots & \dots & 0 & 0\\
1 & 0 & 2 & 0 & 1 & 0 & \dots & \dots & 0 & 0\\
0 & 1 & 0 & 2 & 0 & 1 & \dots & \dots & 0 & 0\\
0 & 0 & 1 & 0 & 2 & 0 & \dots & \dots & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 2 & \dots & \dots & 0 & 0\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & & \vdots & \vdots\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & & \ddots & \vdots & \vdots\\
0 & 0 & 0 & 0 & 0 & 0 & \dots & \dots & 2 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & \dots & \dots & 0 & 1
\end{bmatrix}$$