([b]3[/b]) Let $ \ell$ be the line through $ (0,0)$ and tangent to the curve $ y \equal{} x^3 \plus{} x \plus{} 16$. Find the slope of $ \ell$.

A piece of graph paper is folded once so that $ (0,2)$ is matched with $ (4,0)$ and $ (7,3)$ is matched with $ (m,n)$. Find $ m \plus{} n$. $ \textbf{(A)}\ 6.7\qquad \textbf{(B)}\ 6.8\qquad \textbf{(C)}\ 6.9\qquad \textbf{(D)}\ 7.0\qquad \textbf{(E)}\ 8.0$

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 $OA$.

Externally tangent circles with centers at points $A$ and $B$ have radii of lengths $5$ and $3$, respectively. A line externally tangent to both circles intersects ray $AB$ at point $C$. What is $BC$? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 4.8 \qquad\textbf{(C)}\ 10.2 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 14.4 $

A line passing through $(20,21)$ intersects the curve $y = x^3-2x^2-3x+5$ at three distinct points $A, B,$ and $C,$ such that $B$ is the midpoint of $\overline{AC}$. The slope of this line is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

The graph of the line $ y \equal{} mx \plus{} b$ is shown. Which of the following is true? [asy]import math; unitsize(8mm); defaultpen(linewidth(1pt)+fontsize(6pt)); dashed=linetype("4 4")+linewidth(.8pt); draw((-2,-2.5)--(-2,2.5)--(2.5,2.5)--(2.5,-2.5)--cycle,white); label("$-1$",(-1,0),SW); label("$1$",(1,0),SW); label("$2$",(2,0),SW); label("$1$",(0,1),NE); label("$2$",(0,2),NE); label("$-1$",(0,-1),SW); label("$-2$",(0,-2),SW); drawline((0,0),(1,0)); drawline((0,0),(0,1)); drawline((0,0.8),(1.8,0)); drawline((1,0),(1,1),dashed); drawline((2,0),(2,1),dashed); drawline((-1,0),(-1,1),dashed); drawline((0,1),(1,1),dashed); drawline((0,2),(1,2),dashed); drawline((0,-1),(1,-1),dashed); drawline((0,-2),(1,-2),dashed);[/asy] $ \textbf{(A)}\ mb < \minus{} 1 \qquad \textbf{(B)}\ \minus{} 1 < mb < 0 \qquad \textbf{(C)}\ mb \equal{} 0$ $ \textbf{(D)}\ 0 < mb < 1\qquad \textbf{(E)}\ mb > 1$

Let $P$ be the parabola with equation $y = x^2$ and let $Q = (20, 14)$ There are real numbers $r$ and $s$ such that the line through $Q$ with slope $m$ does not intersect $P$ if and only if $r < m < s$. What is $r + s?$ $ \textbf{(A)} 1 \qquad \textbf{(B)} 26 \qquad \textbf{(C)} 40 \qquad \textbf{(D)} 52 \qquad \textbf{(E)} 80 \qquad $

Let $ \mathcal{C}$ be the hyperbola $ y^2 \minus{} x^2 \equal{} 1$. Given a point $ P_0$ on the $ x$-axis, we construct a sequence of points $ (P_n)$ on the $ x$-axis in the following manner: let $ \ell_n$ be the line with slope $ 1$ passing passing through $ P_n$, then $ P_{n\plus{}1}$ is the orthogonal projection of the point of intersection of $ \ell_n$ and $ \mathcal C$ onto the $ x$-axis. (If $ P_n \equal{} 0$, then the sequence simply terminates.) Let $ N$ be the number of starting positions $ P_0$ on the $ x$-axis such that $ P_0 \equal{} P_{2008}$. Determine the remainder of $ N$ when divided by $ 2008$.

ABC  is a triangle with coordinates A =(2, 6), B =(0, 0), and C =(14, 0). (a) Let P  be the midpoint of AB. Determine the equation of the line perpendicular to AB passing through P. (b) Let Q be the point on line BC  for which PQ is perpendicular to AB. Determine the length of AQ. (c) There is a (unique) circle passing through the points A, B, and C. Determine the radius of this circle.

Let $f(x)$ and $g(x)$ be strictly increasing linear functions from $\mathbb R $ to $\mathbb R $ such that $f(x)$ is an integer if and only if $g(x)$ is an integer. Prove that for any real number $x$, $f(x)-g(x)$ is an integer.

If $r$ is positive and the line whose equation is $x + y = r$ is tangen to the circle whose equation is $x^2 + y ^2 = r$, then $r$ equals $\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }\sqrt{2}\qquad \textbf{(E) }2\sqrt{2}$

Find all the functions $f:[0,1]\rightarrow \mathbb{R}$ for which we have: \[|x-y|^2\le |f(x)-f(y)|\le |x-y|,\] for all $x,y\in [0,1]$.

For which real numbers $c$ is there a straight line that intersects the curve \[ y = x^4 + 9x^3 + cx^2 + 9x + 4\] in four distinct points?

In the following table, each question mark is to be replaced by "Possible" or "Not Possible" to indicate whether a nonvertical line with the given slope can contain the given number of lattice points (points both of whose coordinates are integers). How many of the $12$ entries will be "Possible"? \begin{tabular}{|c|c|c|c|c|} \cline{2-5} \multicolumn{1}{c|}{} & \textbf{zero} & \textbf{exactly one} & \textbf{exactly two} & \textbf{more than two}\\ \hline \textbf{zero slope} & ? & ? & ? & ?\\ \hline \textbf{nonzero rational slope} & ? & ? & ? & ?\\ \hline \textbf{irrational slope} & ? & ? & ? & ?\\ \hline \end{tabular} $ \textbf{(A) }4 \qquad \textbf{(B) }5 \qquad \textbf{(C) }6 \qquad \textbf{(D) }7 \qquad \textbf{(E) }9 \qquad $

In triangle $ABC$, $D$, $E$ and $F$ are the points of tangency of incircle with the center of $I$ to $BC$, $CA$ and $AB$ respectively. Let $M$ be the foot of the perpendicular from $D$ to $EF$. $P$ is on $DM$ such that $DP = MP$. If $H$ is the orthocenter of $BIC$, prove that $PH$ bisects $ EF$.

A parabola has vertex at $(4,-5)$ and has two $x$-intercepts, one positive and one negative. If this parabola is the graph of $y = ax^2 + bx + c$, which of $a$, $b$, and $c$ must be positive? $ \textbf{(A)}\ \text{Only }a\qquad \textbf{(B)}\ \text{Only }b\qquad \textbf{(C)}\ \text{Only }c\qquad \textbf{(D)}\ \text{Only }a\text{ and }b\qquad \textbf{(E)}\ \text{None}$

A car travels due east at $\frac 23$ mile per minute on a long, straight road. At the same time, a circular storm, whose radius is 51 miles, moves southeast at $\frac 12\sqrt{2}$ mile per minute. At time $t=0,$ the center of the storm is 110 miles due north of the car. At time $t=t_1$ minutes, the car enters the storm circle, and at time $t=t_2$ minutes, the car leaves the storm circle. Find $\frac 12(t_1+t_2).$

Find the point in the closed unit disc $D=\{ (x,y) | x^2+y^2\le 1 \}$ at which the function $f(x,y)=x+y$ attains its maximum .

What is the maximum number of rational points that can lie on a circle in $ \mathbb{R}^2$ whose center is not a rational point? (A [i]rational point[/i] is a point both of whose coordinates are rational numbers.)

If the line $L$ in the $xy$-plane has half the slope and twice the y-intercept of the line $y = \frac{2}{3} x + 4$, then an equation for $L$ is: $ \textbf{(A)}\ y = \frac{1}{3} x + 8 \qquad \textbf{(B)}\ y = \frac{4}{3} x + 2 \qquad \textbf{(C)}\ y = \frac{1}{3} x + 4 \qquad\\ \textbf{(D)}\ y = \frac{4}{3} x + 4 \qquad \textbf{(E)}\ y = \frac{1}{3} x + 2 $

Prove that there exists an infinite set of points \[ \dots, \; P_{-3}, \; P_{-2},\; P_{-1},\; P_0,\; P_1,\; P_2,\; P_3,\; \dots \] in the plane with the following property: For any three distinct integers $a,b,$ and $c$, points $P_a$, $P_b$, and $P_c$ are collinear if and only if $a+b+c=2014$.

Let $ABCD$ be a square, and let $E$ and $F$ be points in $AB$ and $BC$ respectively such that $BE=BF$. In the triangle $EBC$, let N be the foot of the altitude relative to $EC$. Let $G$ be the intersection between $AD$ and the extension of the previously mentioned altitude. $FG$ and $EC$ intersect at point $P$, and the lines $NF$ and $DC$ intersect at point $T$. Prove that the line $DP$ is perpendicular to the line $BT$.