Let points $ A = (0,0) , \ B = (1,2), \ C = (3,3), $ and $ D = (4,0) $. Quadrilateral $ ABCD $ is cut into equal area pieces by a line passing through $ A $. This line intersects $ \overline{CD} $ at point $ \left (\frac{p}{q}, \frac{r}{s} \right ) $, where these fractions are in lowest terms. What is $ p + q + r + s $? $ \textbf{(A)} \ 54 \qquad \textbf{(B)} \ 58 \qquad \textbf{(C)} \ 62 \qquad \textbf{(D)} \ 70 \qquad \textbf{(E)} \ 75 $

How many ordered four-tuples of integers $(a,b,c,d)$ with $0 < a < b < c < d < 500$ satisfy $a + d = b + c$ and $bc - ad = 93$?

[color=darkred]Let $n$ and $k$ be two natural numbers such that $n\ge 2$ and $1\le k\le n-1$ . Prove that if the matrix $A\in\mathcal{M}_n(\mathbb{C})$ has exactly $k$ minors of order $n-1$ equal to $0$ , then $\det (A)\ne 0$ .[/color]

Let A,B,C,D four matrices of order n with complex entries, n>=2 and let k real number such that AC+kBD=I and AD=BC. Prove that CA+kDB=I and DA=CB.

Let $A=a_{ij}$ is simetrical real matrix. Prove that : $\sum_i e^{a_{ii}} \leq tr (e^A)$

In any cell of an $n \times n$ table a number is written such that all the rows are distinct. Prove that we can remove a column such that the rows in the new table are still distinct.

Let $p$ be a positive integer, $p>1.$ Find the number of $m\times n$ matrices with entries in the set $\left\{ 1,2,\dots,p\right\} $ and such that the sum of elements on each row and each column is not divisible by $p.$

Let \(A\) be a square matrix with entries in the field \(\mathbb Z / p \mathbb Z\) such that \(A^n - I\) is invertible for every positive integer \(n\). Prove that there exists a positive integer \(m\) such that \(A^m = 0\). [i](A matrix having entries in the field \(\mathbb Z / p \mathbb Z\) means that two matrices are considered the same if each pair of corresponding entries differ by a multiple of \(p\).)[/i] [i]Proposed by Tony Wang[/i]

Hey, This problem is from the VTRMC 2006. 3. Recall that the Fibonacci numbers $ F(n)$ are defined by $ F(0) \equal{} 0$, $ F(1) \equal{} 1$ and $ F(n) \equal{} F(n \minus{} 1) \plus{} F(n \minus{} 2)$ for $ n \geq 2$. Determine the last digit of $ F(2006)$ (e.g. the last digit of 2006 is 6). As, I and a friend were working on this we noticed an interesting relationship when writing the Fibonacci numbers in "mod" notation. Consider the following, 01 = 1 mod 10 01 = 1 mod 10 02 = 2 mod 10 03 = 3 mod 10 05 = 5 mod 10 08 = 6 mod 10 13 = 3 mod 10 21 = 1 mod 10 34 = 4 mod 10 55 = 5 mod 10 89 = 9 mod 10 Now, consider that between the first appearance and second apperance of $ 5 mod 10$, there is a difference of five terms. Following from this we see that the third appearance of $ 5 mod 10$ occurs at a difference 10 terms from the second appearance. Following this pattern we can create the following relationships. $ F(55) \equal{} F(05) \plus{} 5({2}^{2})$ This is pretty much as far as we got, any ideas?

Let be a natural number $ n, $ a number $ t\in (0,1) $ and $ n+1 $ numbers $ a_0\ge a_1\ge a_2\ge\cdots\ge a_n\ge 0. $ Prove the following matrix inequality: $$ \begin{vmatrix}\frac{(1+t\sqrt{-1})^2}{1+t^2} & -1 & 0& 0 & \cdots & 0 & 0 \\ 0 & \frac{(1+t\sqrt{-1})^2}{1+t^2} & -1 & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & \frac{(1+t\sqrt{-1})^2}{1+t^2} & -1 \\ a_0 & a_1 & a_2 & a_3 & \cdots & a_{n-1} & a_n \end{vmatrix}^2\le a_0^2\left( 1+\frac{1}{t^2} \right) $$

Find the derivatives of the lengths of the semiaxes of the ellipsoid $x^2 + y^2 + z^2 + xy + yz + zx = 1 + \epsilon xy$ with respect to $\epsilon$ at $\epsilon = 0$.

Find a matrix $A_{(2 \times 2)}$ for which \[ \begin{bmatrix}2 &1 \\ 3 & 2\end{bmatrix} A \begin{bmatrix}3 & 2 \\ 4 & 3\end{bmatrix} = \begin{bmatrix}1 & 2 \\ 2 & 1\end{bmatrix}.\]

For $n \geq 3$ find the eigenvalues (with their multiplicities) of the $n \times n$ matrix $$\begin{bmatrix} 1 & 0 & 1 & 0 & 0 & 0 & \dots & \dots & 0 & 0\\ 0 & 2 & 0 & 1 & 0 & 0 & \dots & \dots & 0 & 0\\ 1 & 0 & 2 & 0 & 1 & 0 & \dots & \dots & 0 & 0\\ 0 & 1 & 0 & 2 & 0 & 1 & \dots & \dots & 0 & 0\\ 0 & 0 & 1 & 0 & 2 & 0 & \dots & \dots & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 2 & \dots & \dots & 0 & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & & \vdots & \vdots\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & & \ddots & \vdots & \vdots\\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & \dots & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & \dots & 0 & 1 \end{bmatrix}$$

[b]a)[/b] Find all $ 2\times 2 $ complex matrices $ A $ which have the property that there are two complex numbers $ \alpha ,\gamma $ with $ \alpha \neq \text{tr} (A) $ or $ \gamma\neq \det (A) $ such that $ A^2-\alpha A+\gamma I=0. $ [b]b)[/b] Consider $ B\not\in\{ 0,I\} $ as a matrix having the property mentioned at [b]a).[/b] Solve in the complex numbers the system $ xB-yI-B^2=xB^2-yI-B^4=0. $ [i]Adrian Troie[/i]

In the universe of Pi Zone, points are labeled with $2 \times 2$ arrays of positive reals. One can teleport from point $M$ to point $M'$ if $M$ can be obtained from $M'$ by multiplying either a row or column by some positive real. For example, one can teleport from $\left( \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right)$ to $\left( \begin{array}{cc} 1 & 20 \\ 3 & 40 \end{array} \right)$ and then to $\left( \begin{array}{cc} 1 & 20 \\ 6 & 80 \end{array} \right)$. A [i]tourist attraction[/i] is a point where each of the entries of the associated array is either $1$, $2$, $4$, $8$ or $16$. A company wishes to build a hotel on each of several points so that at least one hotel is accessible from every tourist attraction by teleporting, possibly multiple times. What is the minimum number of hotels necessary? [i]Proposed by Michael Kural[/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]

There are two matrices $A$ and $B$ of size $m\times n$ each filled only by “0”s and “1”s. It is given that along any row or column its elements do not decrease (from left to right and from top to bottom).It is also given that the numbers of “1”s in both matrices are equal and for any $k = 1, . . .$ , $m$ the sum of the elements in the top $k$ rows of the matrix $A$ is no less than that of the matrix $B$. Prove for any $l = 1, . . . $, $n$ the sum of the elements in left $l$ columns of the matrix $A$ is no greater than that of the matrix $B$.

The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from $ 1$ to $ 15$ in clockwise order. Committee rules state that a Martian must occupy chair $ 1$ and an Earthling must occupy chair $ 15$. Furthermore, no Earthling can sit immediately to the left of a Martian, no Martian can sit immediately to the left of a Venusian, and no Venusian can sit immediately to the left of an Earthling. The number of possible seating arrangements for the committee is $ N\cdot (5!)^3$. Find $ N$.

In Prime Land, there are seven major cities, labelled $C_0$, $C_1$, \dots, $C_6$. For convenience, we let $C_{n+7} = C_n$ for each $n=0,1,\dots,6$; i.e. we take the indices modulo $7$. Al initially starts at city $C_0$. Each minute for ten minutes, Al flips a fair coin. If the coin land heads, and he is at city $C_k$, he moves to city $C_{2k}$; otherwise he moves to city $C_{2k+1}$. If the probability that Al is back at city $C_0$ after $10$ moves is $\tfrac{m}{1024}$, find $m$. [i]Proposed by Ray Li[/i]

Given a positive integer $n$ and a real valued $n\times n$ matrix $A$. $J$ is $n\times n$ matrix with every entry $1$. Suppose $A$ satisfies the following relations. $$A+A^T = \frac{1}{n} J, \quad AJ = \frac{1}{2} J$$ Show that $A^m-I$ is an invertible matrix for all positive odd integer $m$.

For any square matrix $\mathcal{A}$ we define $\sin {\mathcal{A}}$ by the usual power series. \[ \sin {\mathcal{A}}=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}\mathcal{A}^{2n+1} \] Prove or disprove : $\exists 2\times 2$ matrix $A\in \mathcal{M}_2(\mathbb{R})$ such that \[ \sin{A}=\left(\begin{array}{cc}1 & 1996 \\0 & 1 \end{array}\right) \]

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 $r(j), 1 \leq j \leq d, j\ne i$.

In what case does the system of equations $\begin{matrix} x + y + mz = a \\ x + my + z = b \\ mx + y + z = c \end{matrix}$ have a solution? Find conditions under which the unique solution of the above system is an arithmetic progression.

Consider the $ 4\times 4 $ integer matrices that have the property that each one of them multiplied by its transpose is $ 4I. $ [b]a)[/b] Show that the product of the elements of such a matrix is either $ 0, $ either $ 1. $ [b]b)[/b] How many such matrices have the property that the product of its elements is $ 0? $

Determine all real numbers $a$ such that the two polynomials $x^2+ax+1$ and $x^2+x+a$ have at least one root in common.