Found problems: 823
2003 District Olympiad, 3
a)Prove that any matrix $A\in \mathcal{M}_4(\mathbb{C})$ can be written as a sum of four matrices $B_1,B_2,B_3,B_4\in \mathcal{M}_4(\mathbb{C})$ with the rank equal to $1$.
b)$I_4$ can't be written as a sum of less than four matrices with the rank equal to $1$.
[i]Manuela Prajea & Ion Savu[/i]
2012 European Mathematical Cup, 4
Olja writes down $n$ positive integers $a_1, a_2, \ldots, a_n$ smaller than $p_n$ where $p_n$ denotes the $n$-th prime number. Oleg can choose two (not necessarily different) numbers $x$ and $y$ and replace one of them with their product $xy$. If there are two equal numbers Oleg wins. Can Oleg guarantee a win?
[i]Proposed by Matko Ljulj.[/i]
2005 IMC, 3
What is the maximal dimension of a linear subspace $ V$ of the vector space of real $ n \times n$ matrices such that for all $ A$ in $ B$ in $ V$, we have $ \text{trace}\left(AB\right) \equal{} 0$ ?
1976 IMO Shortlist, 5
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 $j=1, \ldots ,q.$
2024 VJIMC, 2
Let $n$ be a positive integer and let $A$, $B$ be two complex nonsingular $n \times n$ matrices such that
\[A^2B-2ABA+BA^2=0.\]
Prove that the matrix $AB^{-1}A^{-1}B-I_n$ is nilpotent.
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]
2003 VJIMC, Problem 2
Let $ \{D_1, D_2, ..., D_n \}$ be a set of disks in the Euclidean plane. Let $ a_ {i, j} = S (D_i \cap D_j) $ be the area of $ D_i \cap D_j $. Prove that $$ \sum_ {i = 1} ^ n \sum_ {j = 1} ^ n a_ {i, j} x_ix_j \geq 0 $$ for any real numbers $ x_1, x_2, ..., x_n $.
2019 LIMIT Category C, Problem 10
Let $A\in M_3(\mathbb Z)$ such that $\det(A)=1$. What is the maximum possible number of entries of $A$ that are even?
2008 Polish MO Finals, 1
In each cell of a matrix $ n\times n$ a number from a set $ \{1,2,\ldots,n^2\}$ is written --- in the first row numbers $ 1,2,\ldots,n$, in the second $ n\plus{}1,n\plus{}2,\ldots,2n$ and so on. Exactly $ n$ of them have been chosen, no two from the same row or the same column. Let us denote by $ a_i$ a number chosen from row number $ i$. Show that:
\[ \frac{1^2}{a_1}\plus{}\frac{2^2}{a_2}\plus{}\ldots \plus{}\frac{n^2}{a_n}\geq \frac{n\plus{}2}{2}\minus{}\frac{1}{n^2\plus{}1}\]
2008 Alexandru Myller, 2
Let $ A,B,S $ be three $ 3\times 3 $ complex matrices with $ B=S^{-1}AS , $ and $ S $ nonsingular. Show:
$$ \text{tr} \left( B^2\right) +2\text{tr}(C(B)) = \left(\text{tr} (A)\right)^2 , $$
where $ C(B) $ is the cofactor of $ B. $
[i]Mihai Haivas[/i]
2005 Gheorghe Vranceanu, 2
$ 15 $ minors of order $ 3 $ of a $ 4\times 4 $ real matrix whose determinant is a nonzero rational number, are rational.
Prove that this matrix is rational.
MIPT student olimpiad spring 2024, 2
Let the matrix $S$ be orthogonal and the matrix $I-S$ be invertible, where I is the identity
matrix of the same size as $S$.
Find
$x^T(I-S)^{-1}x$
Where $x$ is a real unit vector.
2024 District Olympiad, P1
Consider the matrix $X\in\mathcal{M}_2(\mathbb{C})$ which satisfies $X^{2022}=X^{2023}.$ Prove that $X^2=X^3.$
2010 District Olympiad, 2
Consider the matrix $ A,B\in \mathcal l{M}_3(\mathbb{C})$ with $ A=-^tA$ and $ B=^tB$. Prove that if the polinomial function defined by
\[ f(x)=\det(A+xB)\]
has a multiple root, then $ \det(A+B)=\det B$.
2014 Contests, 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)
2014 SEEMOUS, Problem 1
Let $n$ be a nonzero natural number and $f:\mathbb R\to\mathbb R\setminus\{0\}$ be a function such that $f(2014)=1-f(2013)$. Let $x_1,x_2,x_3,\ldots,x_n$ be real numbers not equal to each other. If
$$\begin{vmatrix}1+f(x_1)&f(x_2)&f(x_3)&\cdots&f(x_n)\\f(x_1)&1+f(x_2)&f(x_3)&\cdots&f(x_n)\\f(x_1)&f(x_2)&1+f(x_3)&\cdots&f(x_n)\\\vdots&\vdots&\vdots&\ddots&\vdots\\f(x_1)&f(x_2)&f(x_3)&\cdots&1+f(x_n)\end{vmatrix}=0,$$prove that $f$ is not continuous.
2008 District Olympiad, 1
If $A\in \mathcal{M}_2(\mathbb{R})$, prove that:
\[\det(A^2+A+I_2)\ge \frac{3}{4}(1-\det A)^2\]
1976 USAMO, 5
If $ P(x),Q(x),R(x)$, and $ S(x)$ are all polynomials such that \[ P(x^5)\plus{}xQ(x^5)\plus{}x^2R(x^5)\equal{}(x^4\plus{}x^3\plus{}x^2\plus{}x\plus{}1)S(x),\] prove that $ x\minus{}1$ is a factor of $ P(x)$.
2004 Nicolae Coculescu, 4
Let be a matrix $ A\in\mathcal{M}_2(\mathbb{R}) $ having the property that the numbers $ \det (A+X) ,\det (A^2+X^2) ,\det (A^3+X^3) $ are (in this order) in geometric progression, for any matrix $ X\in\mathcal{M}_2(\mathbb{R}) . $
Prove that $ A=0. $
[i]Marius Ghergu[/i]
2006 Poland - Second Round, 3
Given is a prime number $p$ and natural $n$ such that $p \geq n \geq 3$. Set $A$ is made of sequences of lenght $n$ with elements from the set $\{0,1,2,...,p-1\}$ and have the following property:
For arbitrary two sequence $(x_1,...,x_n)$ and $(y_1,...,y_n)$ from the set $A$ there exist three different numbers $k,l,m$ such that:
$x_k \not = y_k$, $x_l \not = y_l$, $x_m \not = y_m$.
Find the largest possible cardinality of $A$.
2001 Miklós Schweitzer, 8
Let $H$ be a complex Hilbert space. The bounded linear operator $A$ is called [i]positive[/i] if $\langle Ax, x\rangle \geq 0$ for all $x\in H$. Let $\sqrt A$ be the positive square root of $A$, i.e. the uniquely determined positive operator satisfying $(\sqrt{A})^2=A$. On the set of positive operators we introduce the
$$A\circ B=\sqrt A B\sqrt B$$
operation. Prove that for a given pair $A, B$ of positive operators the identity
$$(A\circ B)\circ C=A\circ (B\circ C)$$
holds for all positive operator $C$ if and only if $AB=BA$.
2012 IMC, 4
Let $n \ge 2$ be an integer. Find all real numbers $a$ such that there exist real numbers $x_1,x_2,\dots,x_n$ satisfying
\[x_1(1-x_2)=x_2(1-x_3)=\dots=x_n(1-x_1)=a.\]
[i]Proposed by Walther Janous and Gerhard Kirchner, Innsbruck.[/i]
2003 Romania National Olympiad, 1
[b]a)[/b] Determine the center of the ring of square matrices of a certain dimensions with elements in a given field, and prove that it is isomorphic with the given field.
[b]b)[/b] Prove that
$$ \left(\mathcal{M}_n\left( \mathbb{R} \right) ,+, \cdot\right)\not\cong \left(\mathcal{M}_n\left( \mathbb{C} \right) ,+,\cdot\right) , $$
for any natural number $ n\ge 2. $
[i]Marian Andronache, Ion Sava[/i]
1984 Putnam, A3
Let $n$ be a positive integer. Let $a,b,x$ be real numbers, with $a \neq b$ and let $M_n$ denote the $2n x 2n $ matrix whose $(i,j)$ entry $m_{ij}$ is given by
$m_{ij}=x$ if $i=j$,
$m_{ij}=a$ if $i \not= j$ and $i+j$ is even,
$m_{ij}=b$ if $i \not= j$ and $i+j$ is odd.
For example
$ M_2=\begin{vmatrix}x& b& a & b\\ b& x & b &a\\ a
& b& x & b\\ b & a & b & x \end{vmatrix}$.
Express $\lim_{x\to\ 0} \frac{ det M_n}{ (x-a)^{(2n-2)} }$ as a polynomial in $a,b $ and $n$ .
P.S. How write in latex $m_{ij}=...$ with symbol for the system (because is multiform function?)
2007 Grigore Moisil Intercounty, 1
Let be two distinct $ 2\times 2 $ real matrices having the property that there exists a natural number such that these matrices raised to this number are equal, and these matrices raised to the successor of this number are also equal.
Prove that these matrices raised to any power greater than $ 2 $ are equal.