A rectangle $ABCD$ has side lengths $AB = m$, $AD = n$, with $m$ and $n$ relatively prime and both odd. It is divided into unit squares and the diagonal AC intersects the sides of the unit squares at the points $A_1 = A, A_2, A_3, \ldots , A_k = C$. Show that \[ A_1A_2 - A_2A_3 + A_3A_4 - \cdots + A_{k-1}A_k = {\sqrt{m^2+n^2}\over mn}. \]

Find all pairs $ (a, b)$ of real numbers such that whenever $ \alpha$ is a root of $ x^{2} \plus{} ax \plus{} b \equal{} 0$, $ \alpha^{2} \minus{} 2$ is also a root of the equation. [b][Weightage 17/100][/b]

Find all triples $(x, y, z)$ such that $x, y, z, x - y, y - z, x - z$ are all prime positive integers.

The following diagram shows an eight-sided polygon $ABCDEFGH$ with side lengths $8,15,8,8,8,6,8,$ and $29$ as shown. All of its angles are right angles. Turn this eight-sided polygon into a six-sided polygon by connecting $B$ to $D$ with an edge and $E$ to $G$ with an edge to form polygon $ABDEGH$. Find the perimeter of $ABDEGH$. [asy] size(200); defaultpen(linewidth(2)); pen qq=font("phvb"); pair rectangle[] = {origin,(0,-8),(15,-8),(15,-16),(23,-16),(23,-8),(29,-8),(29,0)}; string point[] = {"A","B","C","D","E","F","G","H"}; int dirlbl[] = {135,225,225,225,315,315,315,45}; string value[] = {"8","15","8","8","8","6","8","29"}; int direction[] = {0,90,0,90,180,90,180,270}; for(int i=0;i<=7;i=i+1) { draw(rectangle[i]--rectangle[(i+1) % 8]); label(point[i],rectangle[i],dir(dirlbl[i]),qq); label(value[i],(rectangle[i]+rectangle[(i+1) % 8])/2,dir(direction[i]),qq); } [/asy]

Let $ A$ and $ B$ be the endpoints of a semicircular arc of radius $ 2$. The arc is divided into seven congruent arcs by six equally spaced points $ C_1,C_2,\ldots,C_6$. All chords of the form $ \overline{AC_i}$ or $ \overline{BC_i}$ are drawn. Let $ n$ be the product of the lengths of these twelve chords. Find the remainder when $ n$ is divided by $ 1000$.

Let $ n \ge 1$ and $ k \ge 3$ be integers. A circle is divided into $ n$ sectors $ a_1,a_2,\dots,a_n$. We will color the $ n$ sectors with $ k$ different colors such that $ a_i$ and $ a_{i \plus{} 1}$ have different color for each $ i \equal{} 1,2,\dots,n$ where $ a_{n \plus{} 1}\equal{}a_1$. Find the number of ways to do such coloring.

For positive integers $ n$, $ f_n \equal{} \lfloor2^n\sqrt {2008}\rfloor \plus{} \lfloor2^n\sqrt {2009}\rfloor$. Prove there are infinitely many odd numbers and infinitely many even numbers in the sequence $ f_1,f_2,\ldots$.

How many ways are there to line up $19$ girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her?

Rob is helping to build the set for a school play. For one scene, he needs to build a multi-colored tetrahedron out of cloth and bamboo. He begins by fitting three lengths of bamboo together, such that they meet at the same point, and each pair of bamboo rods meet at a right angle. Three more lengths of bamboo are then cut to connect the other ends of the first three rods. Rob then cuts out four triangular pieces of fabric: a blue piece, a red piece, a green piece, and a yellow piece. These triangular pieces of fabric just fill in the triangular spaces between the bamboo, making up the four faces of the tetrahedron. The areas in square feet of the red, yellow, and green pieces are $60$, $20$, and $15$ respectively. If the blue piece is the largest of the four sides, find the number of square feet in its area.

Given 365 cards, in which distinct numbers are written. We may ask for any three cards, the order of numbers written in them. Is it always possible to find out the order of all 365 cards by 2000 such questions?