Found problems: 638
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.
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 \[ \textrm{rank} \left( I_n - A \right) = \textrm{rank} \left( I_n - A^2 \right) = \ldots = \textrm{rank} \left( I_n - A^{p-1} \right) . \]
2004 Italy TST, 1
At the vertices $A, B, C, D, E, F, G, H$ of a cube, $2001, 2002, 2003, 2004, 2005, 2008, 2007$ and $2006$ stones respectively are placed. It is allowed to move a stone from a vertex to each of its three neighbours, or to move a stone to a vertex from each of its three neighbours. Which of the following arrangements of stones at $A, B, \ldots , H$ can be obtained?
$(\text{a})\quad 2001, 2002, 2003, 2004, 2006, 2007, 2008, 2005;$
$(\text{b})\quad 2002, 2003, 2004, 2001, 2006, 2005, 2008, 2007;$
$(\text{c})\quad 2004, 2002, 2003, 2001, 2005, 2008, 2007, 2006.$
2004 Austrian-Polish Competition, 3
Solve the following system of equations in $\mathbb{R}$ where all square roots are non-negative:
$
\begin{matrix}
a - \sqrt{1-b^2} + \sqrt{1-c^2} = d \\
b - \sqrt{1-c^2} + \sqrt{1-d^2} = a \\
c - \sqrt{1-d^2} + \sqrt{1-a^2} = b \\
d - \sqrt{1-a^2} + \sqrt{1-b^2} = c \\
\end{matrix}
$
2016 IMC, 5
Let $A$ be a $n\times n$ complex matrix whose eigenvalues have absolute value at most $1$. Prove that $$ \|A^n\|\le \dfrac{n}{\ln 2} \|A\|^{n-1}. $$ (Here $\|B\|=\sup\limits_{\|x\|\leq 1} \|Bx\|$ for every $n\times n$ matrix $B$ and $\|x\|=\sqrt{\sum\limits_{i=1}^n |x_i|^2}$ for every complex vector $x\in\mathbb{C}^n$.)
(Proposed by Ian Morris and Fedor Petrov, St. Petersburg State University)
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.
1973 Miklós Schweitzer, 1
We say that the rank of a group $ G$ is at most $ r$ if every subgroup of $ G$ can be generated by at most $ r$ elements. Prove
that here exists an integer $ s$ such that for every finite group $ G$ of rank $ 2$ the commutator series of $ G$ has length less than $ s$.
[i]J. Erdos[/i]
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.
2006 Petru Moroșan-Trident, 1
Let be three complex numbers $ \alpha ,\beta ,\gamma $ such that
$$ \begin{vmatrix} \left( \alpha -\beta \right)^2 & \left( \alpha -\beta \right)\left( \beta -\gamma \right) & \left( \beta -\gamma \right)^2 \\ \left( \beta -\gamma \right)^2 & \left( \beta -\gamma \right)\left( \gamma -\alpha \right) & \left( \gamma -\alpha \right)^2 \\ \left( \gamma -\alpha \right)^2 & \left( \gamma -\alpha \right)\left( \alpha -\beta \right) & \left( \alpha -\beta \right)^2\end{vmatrix} =0. $$
Prove that $ \alpha ,\beta ,\gamma $ are all equal, or their affixes represent a non-degenerate equilateral triangle.
[i]Gheorghe Necșuleu[/i] and [i]Ion Necșuleu[/i]
MIPT student olimpiad autumn 2024, 2
$A,B \in M_{2\times 2}(C)$
Prove that:
$Tr(AAABBABAABBB)=tr(BBBAABABBAAA)$
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.
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).
2001 IMC, 5
Let $A$ be an $n\times n$ complex matrix such that $A \ne \lambda I_{n}$ for all $\lambda \in \mathbb{C}$. Prove that $A$ is similar to a matrix having at most one non-zero entry on the maindiagonal.
2008 Putnam, B6
Let $ n$ and $ k$ be positive integers. Say that a permutation $ \sigma$ of $ \{1,2,\dots n\}$ is $ k$-[i]limited[/i] if $ |\sigma(i)\minus{}i|\le k$ for all $ i.$ Prove that the number of $ k$-limited permutations of $ \{1,2,\dots n\}$ is odd if and only if $ n\equiv 0$ or $ 1\pmod{2k\plus{}1}.$
2012 Bogdan Stan, 4
Prove that the elements of any natural power of a $ 2\times 2 $ special linear integer matrix are pairwise coprime, with the possible exception of the pairs that form the diagonals.
[i]Vasile Pop[/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.
1999 IMC, 1
a) Show that $\forall n \in \mathbb{N}_0, \exists A \in \mathbb{R}^{n\times n}: A^3=A+I$.
b) Show that $\det(A)>0, \forall A$ fulfilling the above condition.
1997 Federal Competition For Advanced Students, P2, 1
Let $ a$ be a fixed integer. Find all integer solutions $ x,y,z$ of the system:
$ 5x\plus{}(a\plus{}2)y\plus{}(a\plus{}2)z\equal{}a,$
$ (2a\plus{}4)x\plus{}(a^2\plus{}3)y\plus{}(2a\plus{}2)z\equal{}3a\minus{}1,$
$ (2a\plus{}4)x\plus{}(2a\plus{}2)y\plus{}(a^2\plus{}3)z\equal{}a\plus{}1.$
2010 Iran MO (3rd Round), 5
suppose that $p$ is a prime number. find that smallest $n$ such that there exists a non-abelian group $G$ with $|G|=p^n$.
SL is an acronym for Special Lesson. this year our special lesson was Groups and Symmetries.
the exam time was 5 hours.
2010 CHMMC Winter, 1
A matrix $M$ is called idempotent if $M^2 = M$. Find an idempotent $2 \times 2$ matrix with distinct rational entries or write “none” if none exist.
2014 Contests, 3
Prove that there exists an infinite set of points \[ \dots, \; P_{-3}, \; P_{-2},\; P_{-1},\; P_0,\; P_1,\; P_2,\; P_3,\; \dots \] in the plane with the following property: For any three distinct integers $a,b,$ and $c$, points $P_a$, $P_b$, and $P_c$ are collinear if and only if $a+b+c=2014$.
2008 Rioplatense Mathematical Olympiad, Level 3, 1
In each square of a chessboard with $a$ rows and $b$ columns, a $0$ or $1$ is written satisfying the following conditions.
[list][*]If a row and a column intersect in a square with a $0$, then that row and column have the same number of $0$s.
[*]If a row and a column intersect in a square with a $1$, then that row and column have the same number of $1$s.[/list]
Find all pairs $(a,b)$ for which this is possible.
2002 Romania National Olympiad, 3
Let $A\in M_4(C)$ be a non-zero matrix.
$a)$ If $\text{rank}(A)=r<4$, prove the existence of two invertible matrices $U,V\in M_4(C)$, such that:
\[UAV=\begin{pmatrix}I_r&0\\0&0\end{pmatrix}\]
where $I_r$ is the $r$-unit matrix.
$b)$ Show that if $A$ and $A^2$ have the same rank $k$, then the matrix $A^n$ has rank $k$, for any $n\ge 3$.
2005 iTest, 20
If $A$ is the $3\times 3$ square matrix $\begin{bmatrix}
5 & 3 & 8\\
2 & 2 & 5\\
3 & 5 & 1
\end{bmatrix}$ and $B$ is the $4\times 4$ square matrix $\begin{bmatrix}
32 & 2 & 4 & 3 \\
3 & 4 & 8 & 3 \\
11 & 3 & 6 & 1 \\
5 & 5 & 10 & 1
\end{bmatrix} $ find the sum of the determinants of $A$ and $B$.
1967 IMO Shortlist, 5
Solve the system of equations:
$
\begin{matrix}
x^2 + x - 1 = y \\
y^2 + y - 1 = z \\
z^2 + z - 1 = x.
\end{matrix}
$