Found problems: 638
2014 SEEMOUS, Problem 3
Let $A\in M_n(\mathbb{C}) $ and $a\in \mathbb{C} $ such that $A-A^*=2aI_n $, where $A^*=(\overline{A})^T $ and $I_n$ is identity matrix.
(i) Show that $|\det A|\ge |a|^n $.
(ii) Show that if $|\det A|=|a|^n $ then $A=aI_n$.
2025 Romania National Olympiad, 2
Let $n$ be a positive integer, and $a,b$ be two complex numbers such that $a \neq 1$ and $b^k \neq 1$, for any $k \in \{1,2,\dots ,n\}$. The matrices $A,B \in \mathcal{M}_n(\mathbb{C})$ satisfy the relation $BA=a I_n + bAB$. Prove that $A$ and $B$ are invertible.
2007 Nicolae Coculescu, 1
Let $ \mathbb{K} $ be a field and let be a matrix $ M\in\mathcal{M}_3(\mathbb{K} ) $ having the property that $ \text{tr} (A) =\text{tr} (A^2) =0 . $ Show that there is a $ \mu\in \mathbb{K} $ such that $ A^3=\mu A $ or $ A^3=\mu I. $
[i]Cristinel Mortici[/i]
2006 Victor Vâlcovici, 3
Let be a natural number $ n $ and a matrix $ A\in\mathcal{M}_n(\mathbb{R}) $ having the property that sum of the squares of all its elements is strictly less than $ 1. $ Prove that the matrices $ I\pm A $ are invertible.
2015 Romania National Olympiad, 4
Let be three natural numbers $ k,m,n $ an $ m\times n $ matrix $ A, $ an $ n\times m $ matrix $ B, $ and $ k $ complex numbers $ a_0,a_1,\ldots ,a_k $ such that the following conditions hold.
$ \text{(i)}\quad m\ge n\ge 2 $
$ \text{(ii)}\quad a_0I_m+a_1AB+a_2(AB)^2+\cdots +a_k(AB)^k=O_m $
$ \text{(iii)}\quad a_0I_m+a_1BA+a_2(BA)^2+\cdots +a_k(BA)^k\neq O_n $
Prove that $ a_0=0. $
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 $A^{p}=0_{n}$.
2017 District Olympiad, 3
Let be two matrices $ A,B\in\mathcal{M}_2\left( \mathbb{R} \right) $ that don’t commute.
[b]a)[/b] If $ A^3=B^3, $ then $ \text{tr} \left( A^n \right) =\text{tr} \left( B^n \right) , $ for all natural numbers $ n. $
[b]b)[/b] If $ A^n\neq B^n $ and $ \text{tr} \left( A^n \right) =\text{tr} \left( B^n \right) , $ for all natural numbers $ n, $ then find some of the matrices $ A,B. $
1948 Putnam, B6
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 $0.$
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 $n$.
2005 Alexandru Myller, 1
Let $A,B\in M_2(\mathbb Z)$ s.t. $AB=\begin{pmatrix}1&2005\\0&1\end{pmatrix}$. Prove that there is a matrix $C\in M_2(\mathbb Z)$ s.t. $BA=C^{2005}$.
[i]Dinu Serbanescu[/i]
1997 AMC 12/AHSME, 21
For any positive integer $ n$, let \[f(n) \equal{} \begin{cases} \log_8{n}, & \text{if }\log_8{n}\text{ is rational,} \\
0, & \text{otherwise.} \end{cases}\] What is $ \sum_{n \equal{} 1}^{1997}{f(n)}$?
$ \textbf{(A)}\ \log_8{2047}\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ \frac {55}{3}\qquad \textbf{(D)}\ \frac {58}{3}\qquad \textbf{(E)}\ 585$
2024 Brazil Undergrad MO, 5
Let \( A \) be a \( 2 \times 2 \) matrix with integer entries and \(\det A \neq 0\). If the sequence \(\operatorname{tr}(A^n)\), for \( n = 1, 2, 3, \ldots \), is bounded, show that
\[
A^{12} = I \quad \text{or} \quad (A^2 - I)^2 = O.
\]
Here, \( I \) and \( O \) denote the identity and zero matrices, respectively, and \(\operatorname{tr}\) denotes the trace of the matrix (the sum of the elements on the main diagonal).
1991 Spain Mathematical Olympiad, 2
Given two distinct elements $a,b \in \{-1,0,1\}$, consider the matrix $A$ .
Find a subset $S$ of the set of the rows of $A$, of minimum size, such that every other row of $A$ is a linear combination of the rows in $S$ with integer coefficients.
2011 AMC 12/AHSME, 23
Let $f(z)=\frac{z+a}{z+b}$ and $g(z)=f(f(z))$, where $a$ and $b$ are complex numbers. Suppose that $|a|=1$ and $g(g(z))=z$ for all $z$ for which $g(g(z))$ is defined. What is the difference between the largest and smallest possible values of $|b|$?
$\textbf{(A)}\ 0 \qquad
\textbf{(B)}\ \sqrt{2}-1 \qquad
\textbf{(C)}\ \sqrt{3}-1 \qquad
\textbf{(D)}\ 1 \qquad
\textbf{(E)}\ 2$
2005 Moldova Team Selection Test, 3
Does there exist such a configuration of 22 circles and 22 point, that any circle contains at leats 7 points and any point belongs at least to 7 circles?
2010 China Second Round Olympiad, 4
the code system of a new 'MO lock' is a regular $n$-gon,each vertex labelled a number $0$ or $1$ and coloured red or blue.it is known that for any two adjacent vertices,either their numbers or colours coincide.
find the number of all possible codes(in terms of $n$).
2023 China Team Selection Test, P18
Find the greatest constant $\lambda$ such that for any doubly stochastic matrix of order 100, we can pick $150$ entries such that if the other $9850$ entries were replaced by $0$, the sum of entries in each row and each column is at least $\lambda$.
Note: A doubly stochastic matrix of order $n$ is a $n\times n$ matrix, all entries are nonnegative reals, and the sum of entries in each row and column is equal to 1.
1991 Arnold's Trivium, 3
Find the critical values and critical points of the mapping $z\mapsto z^2+2\overline{z}$ (sketch the answer).
2019 Teodor Topan, 4
Calculate the minimum value of $ \text{tr} (A^tA) , $ where $ A $ in the cases where is a matrix of pairwise distinct nonnegative integers and:
[b]a)[/b] $ \det A\equiv 1\pmod 2 $
[b]b)[/b] $ \det A=0 $
[i]Vlad Mihaly[/i]
2024 AMC 12/AHSME, 14
The numbers, in order, of each row and the numbers, in order, of each column of a $5 \times 5$ array of integers form an arithmetic progression of length $5{.}$ The numbers in positions $(5, 5), \,(2,4),\,(4,3),$ and $(3, 1)$ are $0, 48, 16,$ and $12{,}$ respectively. What number is in position $(1, 2)?$
\[ \begin{bmatrix} . & ? &.&.&. \\ .&.&.&48&.\\ 12&.&.&.&.\\ .&.&16&.&.\\ .&.&.&.&0\end{bmatrix}\]
$\textbf{(A) } 19 \qquad \textbf{(B) } 24 \qquad \textbf{(C) } 29 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 39$
1997 Romania National Olympiad, 1
Let $m \ge 2$ and $n \ge 1$ be integers and $A=(a_{ij})$ a square matrix of order $n$ with integer entries. Prove that for any permutation $\sigma \in S_n$ there is a function $\varepsilon : \{1,2,\ldots,n\} \to \{0,1\}$ such that replacing the entries $a_{\sigma(1)1},$ $a_{\sigma(2)2}, $ $\ldots,$ $a_{\sigma(n)n}$ of $A$ respectively by $$a_{\sigma(1)1}+\varepsilon(1), ~a_{\sigma(2)2}+\varepsilon(2), ~\ldots, ~a_{\sigma(n)n}+\varepsilon(n),$$ the determinant of the matrix $A_{\varepsilon}$ thus obtained is not divisible by $m.$
2004 Germany Team Selection Test, 1
Let n be a positive integer. Find all complex numbers $x_{1}$, $x_{2}$, ..., $x_{n}$ satisfying the following system of equations:
$x_{1}+2x_{2}+...+nx_{n}=0$,
$x_{1}^{2}+2x_{2}^{2}+...+nx_{n}^{2}=0$,
...
$x_{1}^{n}+2x_{2}^{n}+...+nx_{n}^{n}=0$.
2022 IMC, 2
For a positive integer $n$ determine all $n\times n$ real matrices $A$ which have only real eigenvalues and such that there exists an integer $k\geq n$ with $A + A^k = A^T$.
2003 Tournament Of Towns, 7
A $m \times n$ table is filled with signs $"+"$ and $"-"$. A table is called irreducible if one cannot reduce it to the table filled with $"+"$, applying the following operations (as many times as one wishes).
$a)$ It is allowed to flip all the signs in a row or in a column. Prove that an irreducible table contains an irreducible $2\times 2$ sub table.
$b)$ It is allowed to flip all the signs in a row or in a column or on a diagonal (corner cells are diagonals of length $1$). Prove that an irreducible table contains an irreducible $4\times 4$ sub table.
2024 AMC 10, 21
The numbers, in order, of each row and the numbers, in order, of each column of a $5 \times 5$ array of integers form an arithmetic progression of length $5{.}$ The numbers in positions $(5, 5), \,(2,4),\,(4,3),$ and $(3, 1)$ are $0, 48, 16,$ and $12{,}$ respectively. What number is in position $(1, 2)?$
\[ \begin{bmatrix} . & ? &.&.&. \\ .&.&.&48&.\\ 12&.&.&.&.\\ .&.&16&.&.\\ .&.&.&.&0\end{bmatrix}\]
$\textbf{(A) } 19 \qquad \textbf{(B) } 24 \qquad \textbf{(C) } 29 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 39$