Found problems: 638
A triangle is bounded by the lines $a_1 x+ b_1 y +c_1=0$, $a_2 x+ b_2 y +c_2=0$ and $a_2 x+ b_2 y +c_2=0$.
Show that its area, disregarding sign, is
$$\frac{\Delta^{2}}{2(a_2 b_3- a_3 b_2)(a_3 b_1- a_1 b_3)(a_1 b_2- a_2 b_1)},$$
where $\Delta$ is the discriminant of the matrix
$$M=\begin{pmatrix}
a_1 & b_1 &c_1\\
a_2 & b_2 &c_2\\
a_3 & b_3 &c_3
\end{pmatrix}.$$
Let $n \ge 2$ be an integer, and let $O$ be the $n \times n$ matrix whose entries are all equal to $0$. Two distinct entries of the matrix are chosen uniformly at random, and those two entries are changed from $0$ to $1$. Call the resulting matrix $A$.
Determine the probability that $A^2 = O$, as a function of $n$.
Let $\mathcal{M}$ be the set of $5\times 5$ real matrices of rank $3$. Given a matrix in $\mathcal{M}$, the set of columns of $A$ has $2^5-1=31$ nonempty subsets. Let $k_A$ be the number of these subsets that are linearly independent.
Determine the maximum and minimum values of $k_A$, as $A$ varies over $\mathcal{M}$.
[i]The rank of a matrix is the dimension of the span of its columns.[/i]
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.
Let be a natural number $ n\ge 2. $ Prove that there exist exactly two subsets of the set $ \left\{ \left.\left(\begin{matrix} a& b\\-b& a \end{matrix}\right)\right| a,b\in\mathbb{R} \right\} $ that are closed under multiplication and their cardinal is $ n. $
[i]Marcel Tena[/i]
Let n be a positive integer. Find all complex numbers $x_{1}$, $x_{2}$, ..., $x_{n}$ satisfying the following system of equations:
$x_{1}+2x_{2}+...+nx_{n}=0$,
$x_{1}^{2}+2x_{2}^{2}+...+nx_{n}^{2}=0$,
...
$x_{1}^{n}+2x_{2}^{n}+...+nx_{n}^{n}=0$.
Solve the system of equations:
$
\begin{matrix}
x^2 + x - 1 = y \\
y^2 + y - 1 = z \\
z^2 + z - 1 = x.
\end{matrix}
$
For an ${n\times n}$ matrix $A$, let $X_{i}$ be the set of entries in row $i$, and $Y_{j}$ the set of entries in column $j$, ${1\leq i,j\leq n}$. We say that $A$ is [i]golden[/i] if ${X_{1},\dots ,X_{n},Y_{1},\dots ,Y_{n}}$ are distinct sets. Find the least integer $n$ such that there exists a ${2004\times 2004}$ golden matrix with entries in the set ${\{1,2,\dots ,n\}}$.
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 $n\ge 3$.
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 $(1, 2)?$
\[ \begin{bmatrix} . & ? &.&.&. \\ .&.&.&48&.\\ 12&.&.&.&.\\ .&.&16&.&.\\ .&.&.&.&0\end{bmatrix}\]
$\textbf{(A) } 19 \qquad \textbf{(B) } 24 \qquad \textbf{(C) } 29 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 39$
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?
Ivan writes the matrix $\begin{pmatrix} 2 & 3\\ 2 & 4\end{pmatrix}$ on the board. Then he performs the following operation on the matrix several times:
[b]1.[/b] he chooses a row or column of the matrix, and
[b]2.[/b] he multiplies or divides the chosen row or column entry-wise by the other row or column, respectively.
Can Ivan end up with the matrix $\begin{pmatrix} 2 & 4\\ 2 & 3\end{pmatrix}$ after finitely many steps?
Let $G= \{ A \in \mathcal M_2 \left( \mathbb C \right) \mid |\det A| = 1 \}$ and $H =\{A \in \mathcal M_2 \left( \mathbb C \right) \mid \det A = 1 \}$. Prove that $G$ and $H$ together with the operation of matrix multiplication are two non-isomorphical groups.
For a matrix $(p_{ij})$ of the format $m\times n$ with real entries, set
\[a_i =\displaystyle\sum_{j=1}^n p_{ij}\text{ for }i = 1,\cdots,m\text{ and }b_j =\displaystyle\sum_{i=1}^m p_{ij}\text{ for }j = 1, . . . , n\longrightarrow(1)\]
By integering a real number, we mean replacing the number with the integer closest to it. Prove that integering the numbers $a_i, b_j, p_{ij}$ can be done in such a way that $(1)$ still holds.
For which positive integers $n$ is there an $n\times n$ matrix with integer entries such that every dot product of a row with itself is even, while every dot product of two different rows is odd?
A cake has the form of an $ n$ x $ n$ square composed of $ n^{2}$ unit squares. Strawberries lie on some of the unit squares so that each row or column contains exactly one strawberry; call this arrangement $\mathcal{A}$.
Let $\mathcal{B}$ be another such arrangement. Suppose that every grid rectangle with one vertex at the top left corner of the cake contains no fewer strawberries of arrangement $\mathcal{B}$ than of arrangement $\mathcal{A}$. Prove that arrangement $\mathcal{B}$ can be obtained from $ \mathcal{A}$ by performing a number of switches, defined as follows:
A switch consists in selecting a grid rectangle with only two strawberries, situated at its top right corner and bottom left corner, and moving these two strawberries to the other two corners of that rectangle.
What is the biggest shadow that a cube of side length $1$ can have, with the sun at its peak?
Note: "The biggest shadow of a figure with the sun at its peak" is understood to be the biggest possible area of the orthogonal projection of the figure on a plane.
For an ${n\times n}$ matrix $A$, let $X_{i}$ be the set of entries in row $i$, and $Y_{j}$ the set of entries in column $j$, ${1\leq i,j\leq n}$. We say that $A$ is [i]golden[/i] if ${X_{1},\dots ,X_{n},Y_{1},\dots ,Y_{n}}$ are distinct sets. Find the least integer $n$ such that there exists a ${2004\times 2004}$ golden matrix with entries in the set ${\{1,2,\dots ,n\}}$.
Sequences $(x_n)_{n\ge1}$, $(y_n)_{n\ge1}$ satisfy the relations $x_n=4x_{n-1}+3y_{n-1}$ and $y_n=2x_{n-1}+3y_{n-1}$ for $n\ge1$. If $x_1=y_1=5$ find $x_n$ and $y_n$.
Calculate $\lim_{n\rightarrow\infty}\frac{x_n}{y_n}$.
Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.
Let $d\leq n$ be positive integers and $A$ a real $d\times n$ matrix. Let $\sigma(A)$ be the supremum of $\inf_{v\in W,|v|=1}|Av|$ over all subspaces $W$ of $R^n$ with dimension $d$.
For each $j \leq d$, let $r(j) \in \mathbb{R}^n$ be the $j$th row vector of $A$. Show that:
\[\sigma(A) \leq \min_{i\leq d} d(r(i), \langle r(j), j\ne i\rangle) \leq \sqrt{n}\sigma(A)\]
In which all are euclidian norms and $d(r(i), \langle r(j), j\ne i\rangle)$ denotes the distance between $r(i)$ and the span of $r(j), 1 \leq j \leq d, j\ne i$.
2500 chess kings have to be placed on a $100 \times 100$ chessboard so that
[b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex);
[b](ii)[/b] each row and each column contains exactly 25 kings.
Find the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.)
[i]Proposed by Sergei Berlov, Russia[/i]
Does there exist a real $3\times 3$ matrix $A$ such that $\text{tr}(A)=0$ and $A^2+A^t=I?$ ($\text{tr}(A)$ denotes the trace of $A,\ A^t$ the transpose of $A,$ and $I$ is the identity matrix.)
[i]Proposed by Moubinool Omarjee, Paris[/i]
Let $A\in\mathcal{M}_n(\mathbb{C})$ such that $A^*A^2 = AA^*$. Prove that $A^2=A$. (Here we denote by $A^*$ the conjugate transpose of $A$.)
Let $A$ be an $n\times n$ real matrix with minimal polynomial $x^n + x - 1$. Prove that the trace of $(nA^{n-1} + I)^{-1}A^{n-2}$ is zero.