You plot weight $(y)$ against height $(x)$ for three of your friends and obtain the points $(x_{1},y_{1})$, $(x_{2},y_{2})$, $(x_{3},y_{3})$. If \[x_{1} < x_{2} < x_{3}\quad\text{ and }\quad x_{3} - x_{2} = x_{2} - x_{1},\] which of the following is necessarily the slope of the line which best fits the data? "Best fits" means that the sum of the squares of the vertical distances from the data points to the line is smaller than for any other line. $ \textbf{(A)}\ \frac{y_{3} - y_{1}}{x_{3} - x_{1}}\qquad\textbf{(B)}\ \frac{(y_{2} - y_{1}) - (y_{3} - y_{2})}{x_{3} - x_{1}}\qquad\textbf{(C)}\ \frac{2y_{3} - y_{1} - y_{2}}{2x_{3} - x_{1} - x_{2}}\qquad\textbf{(D)}\ \frac{y_{2} - y_{1}}{x_{2} - x_{1}} + \frac{y_{3} - y_{2}}{x_{3} - x_{2}}\qquad\textbf{(E)}\ \text{none of these} $

A point $(x,y)$ in the first quadrant lies on a line with intercepts $(a,0)$ and $(0,b)$, with $a,b > 0$. Rectangle $M$ has vertices $(0,0)$, $(x,0)$, $(x,y)$, and $(0,y)$, while rectangle $N$ has vertices $(x,y)$, $(x,b)$, $(a,b)$, and $(a,y)$. What is the ratio of the area of $M$ to that of $N$? [i]Proposed by Eugene Chen[/i]

In rectangle $ ABCD$, we have $ A \equal{} (6, \minus{} 22)$, $ B \equal{} (2006,178)$, and $ D \equal{} (8,y)$, for some integer $ y$. What is the area of rectangle $ ABCD$? $ \textbf{(A) } 4000 \qquad \textbf{(B) } 4040 \qquad \textbf{(C) } 4400 \qquad \textbf{(D) } 40,000 \qquad \textbf{(E) } 40,400$

Let $\mathcal{H}$ be the region of points $(x, y)$, such that $(1, 0), (x, y), (-x, y)$, and $(-1,0)$ form an isosceles trapezoid whose legs are shorter than the base between $(x, y)$ and $(-x,y)$. Find the least possible positive slope that a line could have without intersecting $\mathcal{H}$.

The $y$-intercepts, $P$ and $Q$, of two perpendicular lines intersecting at the point $A(6,8)$ have a sum of zero. What is the area of $\triangle APQ$? $ \textbf{(A)}\ 45\qquad\textbf{(B)}\ 48\qquad\textbf{(C)}\ 54\qquad\textbf{(D)}\ 60\qquad\textbf{(E)}\ 72 $

Let $ABC$ be an acute-angled triangle with $AB<AC$ and let $O$ be the centre of its circumcircle $\omega$. Let $D$ be a point on the line segment $BC$ such that $\angle BAD = \angle CAO$. Let $E$ be the second point of intersection of $\omega$ and the line $AD$. If $M$, $N$ and $P$ are the midpoints of the line segments $BE$, $OD$ and $AC$, respectively, show that the points $M$, $N$ and $P$ are collinear.

Let $ ABCDE$ be a convex pentagon. If $ P$, $ Q$, $ R$ and $ S$ are the respective centroids of the triangles $ ABE$, $ BCE$, $ CDE$ and $ DAE$, show that $ PQRS$ is a parallelogram and its area is $ 2/9$ of that of $ ABCD$.

A large wedge rests on a horizontal frictionless surface, as shown. A block starts from rest and slides down the inclined surface of the wedge, which is rough. During the motion of the block, the center of mass of the block and wedge [asy] draw((0,0)--(10,0),linewidth(1)); filldraw((2.5,0)--(6.5,2.5)--(6.5,0)--cycle, gray(.9),linewidth(1)); filldraw((5, 12.5/8)--(6,17.5/8)--(6-5/8, 17.5/8+1)--(5-5/8,12.5/8+1)--cycle, gray(.2)); [/asy] $\textbf{(A)}\ \text{does not move}$ $\textbf{(B)}\ \text{moves horizontally with constant speed}$ $\textbf{(C)}\ \text{moves horizontally with increasing speed}$ $\textbf{(D)}\ \text{moves vertically with increasing speed}$ $\textbf{(E)}\ \text{moves both horizontally and vertically}$

In the future city Baltic Way there are sixteen hospitals. Every night exactly four of them must be on duty for emergencies. Is it possible to arrange the schedule in such a way that after twenty nights every pair of hospitals have been on common duty exactly once?

Let the major axis of an ellipse be $AB$, let $O$ be its center, and let $F$ be one of its foci. $P$ is a point on the ellipse, and $CD$ a chord through $O$, such that $CD$ is parallel to the tangent of the ellipse at $P$. $PF$ and $CD$ intersect at $Q$. Compare the lengths of $PQ$ and $OA$.

Two nonhorizontal, non vertical lines in the $xy$-coordinate plane intersect to form a $45^{\circ}$ angle. One line has slope equal to $6$ times the slope of the other line. What is the greatest possible value of the product of the slopes of the two lines? $\textbf{(A)}\ \frac16 \qquad\textbf{(B)}\ \frac23 \qquad\textbf{(C)}\ \frac32 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 6$

In a 10 by 10 grid of dots, what is the maximum number of lines that can be drawn connecting two dots on the grid so that no two lines are parallel? [i]Proposed by Aaron Lin[/i]

Let $ABC$ be an acute-angled triangle with $AB<AC$ and let $O$ be the centre of its circumcircle $\omega$. Let $D$ be a point on the line segment $BC$ such that $\angle BAD = \angle CAO$. Let $E$ be the second point of intersection of $\omega$ and the line $AD$. If $M$, $N$ and $P$ are the midpoints of the line segments $BE$, $OD$ and $AC$, respectively, show that the points $M$, $N$ and $P$ are collinear.

Let $ABC$ be a triangle with $|AB|=|AC|$. $D$ is the midpoint of $[BC]$. $E$ is the foot of the altitude from $D$ to $AC$. $BE$ cuts the circumcircle of triangle $ABD$ at $B$ and $F$. $DE$ and $AF$ meet at $G$. Prove that $|DG|=|GE|$