Found problems: 823
2016 District Olympiad, 2
Show that:
$$ 2015\in\left\{ x_1+2x_2+3x_3\cdots +2015x_{2015}\big| x_1,x_2,\ldots ,x_{2015}\in \{ -2,3\}\right\}\not\ni 2016. $$
2010 Singapore MO Open, 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.
2019 IMC, 9
Determine all positive integers $n$ for which there exist $n\times n$ real invertible matrices $A$ and $B$ that satisfy $AB-BA=B^2A$.
[i]Proposed by Karen Keryan, Yerevan State University & American University of Armenia, Yerevan[/i]
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 $|b|$?
$\textbf{(A)}\ 0 \qquad
\textbf{(B)}\ \sqrt{2}-1 \qquad
\textbf{(C)}\ \sqrt{3}-1 \qquad
\textbf{(D)}\ 1 \qquad
\textbf{(E)}\ 2$
1988 Greece National Olympiad, 3
Let $A$ be a $n \times n$ matrix of real numbers such that $A^2+\mathbb{I}=A$, where $\mathbb{I}$ is the identity $n \times n$ matrix. Prove that the matrix $A^{3n}$ , where $\nu\in\mathbb{Z}$ takes only two values and find those values.
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.
2005 Brazil Undergrad MO, 6
Prove that for any natural numbers $0 \leq i_1 < i_2 < \cdots < i_k$ and $0 \leq j_1 < j_2 < \cdots < j_k$, the matrix $A = (a_{rs})_{1\leq r,s\leq k}$, $a_{rs} = {i_r + j_s\choose i_r} = {(i_r + j_s)!\over i_r!\, j_s!}$ ($1\leq r,s\leq k$) is nonsingular.
2002 China Western Mathematical Olympiad, 4
Let $ n$ be a positive integer, let the sets $ A_{1},A_{2},\cdots,A_{n \plus{} 1}$ be non-empty subsets of the set $ \{1,2,\cdots,n\}.$ prove that there exist two disjoint non-empty subsets of the set $ \{1,2,\cdots,n \plus{} 1\}$: $ \{i_{1},i_{2},\cdots,i_{k}\}$ and $ \{j_{1},j_{2},\cdots,j_{m}\}$ such that $ A_{i_{1}}\cup A_{i_{2}}\cup\cdots\cup A_{i_{k}} \equal{} A_{j_{1}}\cup A_{j_{2}}\cup\cdots\cup A_{j_{m}}$.
2002 IMC, 7
Compute the determinant of the $n\times n$ matrix $A=(a_{ij})_{ij}$,
$$a_{ij}=\begin{cases}
(-1)^{|i-j|} & \text{if}\, i\ne j,\\
2 & \text{if}\, i= j.
\end{cases}$$
2018 Romania National Olympiad, 4
Let $n$ be an integer with $n \geq 2$ and let $A \in \mathcal{M}_n(\mathbb{C})$ such that $\operatorname{rank} A \neq \operatorname{rank} A^2.$ Prove that there exists a nonzero matrix $B \in \mathcal{M}_n(\mathbb{C})$ such that $$AB=BA=B^2=0$$
[i]Cornel Delasava[/i]
2021 IMC, 5
Let $A$ be a real $n \times n$ matrix and suppose that for every positive integer $m$ there exists a real symmetric matrix $B$ such that
$$2021B = A^m+B^2.$$
Prove that $|\text{det} A| \leq 1$.
2014 USA TSTST, 4
Let $P(x)$ and $Q(x)$ be arbitrary polynomials with real coefficients, and let $d$ be the degree of $P(x)$. Assume that $P(x)$ is not the zero polynomial. Prove that there exist polynomials $A(x)$ and $B(x)$ such that:
(i) both $A$ and $B$ have degree at most $d/2$
(ii) at most one of $A$ and $B$ is the zero polynomial.
(iii) $\frac{A(x)+Q(x)B(x)}{P(x)}$ is a polynomial with real coefficients. That is, there is some polynomial $C(x)$ with real coefficients such that $A(x)+Q(x)B(x)=P(x)C(x)$.
2007 IMS, 4
Prove that: \[\det(A)=\frac{1}{n!}\left| \begin{array}{llllll}\mbox{tr}(A) & 1 & 0 & \ldots & \ldots & 0 \\ \mbox{tr}(A^{2}) & \mbox{tr}(A) & 2 & 0 & \ldots & 0 \\ \mbox{tr}(A^{3}) & \mbox{tr}(A^{2}) & \mbox{tr}(A) & 3 & & \vdots \\ \vdots & & & & & n-1 \\ \mbox{tr}(A^{n}) & \mbox{tr}(A^{n-1}) & \mbox{tr}(A^{n-2}) & \ldots & \ldots & \mbox{tr}(A) \end{array}\right|\]
2019 LIMIT Category C, Problem 1
Which of the following are true?
$\textbf{(A)}~\forall A\in M_n(\mathbb R),A^t=X^{-1}AX\text{ for some }X\in M_n(\mathbb R)$
$\textbf{(B)}~\forall A\in M_n(\mathbb R),I+AA^t\text{ is invertible}$
$\textbf{(C)}~\operatorname{tr}(AB)=\operatorname{tr}(BA),\forall A,B\in M_n(\mathbb R)\text{ but }\exists A,B,C\text{ such that }\operatorname{tr}(ABC)\ne\operatorname{tr}(BAC)$
$\textbf{(D)}~\text{None of the above}$
2008 Harvard-MIT Mathematics Tournament, 29
Let $ (x,y)$ be a pair of real numbers satisfying \[ 56x \plus{} 33y \equal{} \frac{\minus{}y}{x^2\plus{}y^2}, \qquad \text{and} \qquad 33x\minus{}56y \equal{} \frac{x}{x^2\plus{}y^2}.
\]Determine the value of $ |x| \plus{} |y|$.
2016 Romania National Olympiad, 2
Consider a natural number, $ n\ge 2, $ and three $ n\times n $ complex matrices $ A,B,C $ such that $ A $ is invertible, $ B $ is formed by replacing the first line of $ A $ with zeroes, and $ C $ is formed by putting the last $ n-1 $ lines of $ A $ above a line of zeroes. Prove that:
[b]a)[/b] $ \text{rank} \left( A^{-1} B \right) = \text{rank} \left( \left( A^{-1} B\right)^2 \right) =\cdots =\text{rank} \left( \left( A^{-1} B\right)^n \right) $
[b]b)[/b] $ \text{rank} \left( A^{-1} C \right) > \text{rank} \left( \left( A^{-1} C\right)^2 \right) >\cdots >\text{rank} \left( \left( A^{-1} C\right)^n \right) $
1947 Putnam, A6
A $3\times 3$ matrix has determinant $0$ and the cofactor of any element is equal to the square of that element. Show that every element in the matrix is $0.$
1995 Putnam, 5
Let $x_1,x_2,\cdots, x_n$ be real valued differentiable functions of a variable $t$ which satisfy
\begin{align*}
& \frac{\mathrm{d}x_1}{\mathrm{d}t}=a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n\\
& \frac{\mathrm{d}x_2}{\mathrm{d}t}=a_{21}x_1+a_{22}x_2+\cdots+a_{2n}x_n\\
& \;\qquad \vdots \\
& \frac{\mathrm{d}x_n}{\mathrm{d}t}=a_{n1}x_1+a_{n2}x_2+\cdots+a_{nn}x_n\\
\end{align*}
For some constants $a_{ij}>0$. Suppose that $\lim_{t \to \infty}x_i(t)=0$ for all $1\le i \le n$. Are the functions $x_i$ necessarily linearly dependent?
2010 Contests, 2
How many ordered pairs of positive integers $(x,y)$ are there such that $y^2-x^2=2y+7x+4$?
$ \textbf{(A)}\ 3
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 1
\qquad\textbf{(D)}\ 0
\qquad\textbf{(E)}\ \text{Infinitely many}
$
1997 Romania National Olympiad, 2
Let $A$ be a square matrix of odd order (at least $3$) whose entries are odd integers. Prove that if $A$ is invertible, then it is not possible for all the minors of the entries of a row to have equal absolute values.
2012 Bogdan Stan, 1
Let be two $ 2\times 2 $ real matrices $A,B$ having the property that all their natural powers are not real multiples of the identity. Prove that if some natural power of $ A $ is equal to some natural power of $ B, $ then, $ A,B $ commute. Is the converse statement true?
[i]Cosmin Nitu[/i]
1996 Turkey Team Selection Test, 2
Find the maximum number of pairwise disjoint sets of the form
$S_{a,b} = \{n^{2}+an+b | n \in \mathbb{Z}\}$, $a, b \in \mathbb{Z}$.
2011 Romania Team Selection Test, 2
Given a prime number $p$ congruent to $1$ modulo $5$ such that $2p+1$ is also prime, show that there exists a matrix of $0$s and $1$s containing exactly $4p$ (respectively, $4p+2$) $1$s no sub-matrix of which contains exactly $2p$ (respectively, $2p+1$) $1$s.
2007 ITAMO, 4
Today is Barbara's birthday, and Alberto wants to give her a gift playing the following game. The numbers 0,1,2,...,1024 are written on a blackboard. First Barbara erases $2^{9}$ numbers, then Alberto erases $2^{8}$ numbers, then Barbara $2^{7}$ and so on, until there are only two numbers a,b left. Now Barbara earns $|a-b|$ euro.
Find the maximum number of euro that Barbara can always win, independently of Alberto's strategy.
1985 Miklós Schweitzer, 11
Let $\xi (E, \pi, B)\, (\pi\colon E\rightarrow B)$ be a real vector bundle of finite rank, and let
$$\tau_E=V\xi \oplus H\xi\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (*)$$
be the tangent bundle of $E$, where $V\xi=\mathrm{Ker}\, d\pi$ is the vertical subbundle of $\tau_E$. Let us denote the projection operators corresponding to the splitting $(*)$ by $v$ and $h$. Construct a linear connection $\nabla$ on $V\xi$ such that
$$\nabla_X\lor Y - \nabla_Y \lor X=v[X,Y] - v[hX,hY]$$
($X$ and $Y$ are vector fields on $E$, $[.,\, .]$ is the Lie bracket, and all data are of class $\mathcal C^\infty$. [J. Szilasi]