A line with slope $2$ intersects a line with slope $6$ at the point $(40, 30)$. What is the distance between the $x$-intercepts of these two lines? $\textbf{(A) }5\qquad\textbf{(B) }10\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }50$

Circles with centers $ (2,4)$ and $ (14,9)$ have radii 4 and 9, respectively. The equation of a common external tangent to the circles can be written in the form $ y \equal{} mx \plus{} b$ with $ m > 0$. What is $ b$? [asy] size(150); defaultpen(linewidth(0.7)+fontsize(8)); draw(circle((2,4),4));draw(circle((14,9),9)); draw((0,-2)--(0,20));draw((-6,0)--(25,0)); draw((2,4)--(2,4)+4*expi(pi*4.5/11)); draw((14,9)--(14,9)+9*expi(pi*6/7)); label("4",(2,4)+2*expi(pi*4.5/11),(-1,0)); label("9",(14,9)+4.5*expi(pi*6/7),(1,1)); label("(2,4)",(2,4),(0.5,-1.5));label("(14,9)",(14,9),(1,-1)); draw((-4,120*-4/119+912/119)--(11,120*11/119+912/119)); dot((2,4)^^(14,9));[/asy] $ \textbf{(A) } \frac {908}{199}\qquad \textbf{(B) } \frac {909}{119}\qquad \textbf{(C) } \frac {130}{17}\qquad \textbf{(D) } \frac {911}{119}\qquad \textbf{(E) } \frac {912}{119}$

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] $\textbf{(A)}\ \text{does not move}$ $\textbf{(B)}\ \text{moves horizontally with constant speed}$ $\textbf{(C)}\ \text{moves horizontally with increasing speed}$ $\textbf{(D)}\ \text{moves vertically with increasing speed}$ $\textbf{(E)}\ \text{moves both horizontally and vertically}$

Tess runs counterclockwise around rectangular block JKLM. She lives at corner J. Which graph could represent her straight-line distance from home? [asy]pair J=(0,6), K=origin, L=(10,0), M=(10,6); draw(J--K--L--M--cycle); label("$J$", J, dir((5,3)--J)); label("$K$", K, dir((5,3)--K)); label("$L$", L, dir((5,3)--L)); label("$M$", M, dir((5,3)--M));[/asy] $\textbf{(A)}$ [asy]size(80);defaultpen(linewidth(0.8)); draw((16,0)--origin--(0,16)); draw(origin--(15,15)); label("time", (8,0), S); label(rotate(90)*"distance", (0,8), W); [/asy] $\textbf{(B)}$ [asy]size(80);defaultpen(linewidth(0.8)); draw((16,0)--origin--(0,16)); draw((0,6)--(1,6)--(1,12)--(2,12)--(2,11)--(3,11)--(3,1)--(12,1)--(12,0)); label("time", (8,0), S); label(rotate(90)*"distance", (0,8), W); [/asy] $\textbf{(C)}$ [asy]size(80);defaultpen(linewidth(0.8)); draw((16,0)--origin--(0,16)); draw(origin--(2.7,8)--(3,9)^^(11,9)--(14,0)); draw(Arc((4,9), 1, 0, 180)); draw(Arc((10,9), 1, 0, 180)); draw(Arc((7,9), 2, 180,360)); label("time", (8,0), S); label(rotate(90)*"distance", (0,8), W); [/asy] $\textbf{(D)}$ [asy]size(80);defaultpen(linewidth(0.8)); draw((16,0)--origin--(0,16)); draw(origin--(2,6)--(7,14)--(10,12)--(14,0)); label("time", (8,0), S); label(rotate(90)*"distance", (0,8), W); [/asy] $\textbf{(E)}$ [asy]size(80);defaultpen(linewidth(0.8)); draw((16,0)--origin--(0,16)); draw(origin--(3,6)--(7,6)--(10,12)--(14,12)); label("time", (8,0), S); label(rotate(90)*"distance", (0,8), W); [/asy]

Find the positive value of $ a$ such that the curve $ C_1: x \equal{} \sqrt {2y^2 \plus{} \frac {25}{2}}$ tangent to the parabola $ C_2: y \equal{} ax^2$, then find the equation of the tangent line of $ C_1$ at the point of tangency.

Circles $\mathcal{C}_1$, $\mathcal{C}_2$, and $\mathcal{C}_3$ have their centers at (0,0), (12,0), and (24,0), and have radii 1, 2, and 4, respectively. Line $t_1$ is a common internal tangent to $\mathcal{C}_1$ and $\mathcal{C}_2$ and has a positive slope, and line $t_2$ is a common internal tangent to $\mathcal{C}_2$ and $\mathcal{C}_3$ and has a negative slope. Given that lines $t_1$ and $t_2$ intersect at $(x,y)$, and that $x=p-q\sqrt{r}$, where $p$, $q$, and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r$.

What is the minimum value of $ f(x) \equal{} |x \minus{} 1| \plus{} |2x \minus{} 1| \plus{} |3x \minus{} 1| \plus{} \cdots \plus{} |119x \minus{} 1|$? $ \textbf{(A)}\ 49 \qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 51 \qquad \textbf{(D)}\ 52 \qquad \textbf{(E)}\ 53$

In the usual coordinate system $ xOy$ a line $ d$ intersect the circles $ C_1:$ $ (x\plus{}1)^2\plus{}y^2\equal{}1$ and $ C_2:$ $ (x\minus{}2)^2\plus{}y^2\equal{}4$ in the points $ A,B,C$ and $ D$ (in this order). It is known that $ A\left(\minus{}\frac32,\frac{\sqrt3}2\right)$ and $ \angle{BOC}\equal{}60^{\circ}$. All the $ Oy$ coordinates of these $ 4$ points are positive. Find the slope of $ d$.

$56$ lines are drawn on a plane such that no three of them are concurrent. If the lines intersect at exactly $594$ points, what is the maximum number of them that could have the same slope?

The number of points common to the graphs of \[(x-y+2)(3x+y-4)=0\text{ and }(x+y-2)(2x-5y+7)=0\] is: $\textbf{(A) }2\qquad \textbf{(B) }4\qquad \textbf{(C) }6\qquad \textbf{(D) }16\qquad \textbf{(E) }\text{infinite}$

Sketch the graph of the derivative and the graph of the integral of a function given by a free-hand graph.

Let $f$ be a differentiable real-valued function defined on the positive real numbers. The tangent lines to the graph of $f$ always meet the $y$-axis 1 unit lower than where they meet the function. If $f(1)=0$, what is $f(2)$?

Positive integers $ a$, $ b$, and $ c$ are chosen so that $ a<b<c$, and the system of equations \[ 2x\plus{}y\equal{}2003\text{ and }y\equal{}|x\minus{}a|\plus{}|x\minus{}b|\plus{}|x\minus{}c| \]has exactly one solution. What is the minimum value of $ c$? $ \textbf{(A)}\ 668 \qquad \textbf{(B)}\ 669 \qquad \textbf{(C)}\ 1002 \qquad \textbf{(D)}\ 2003 \qquad \textbf{(E)}\ 2004$

For the simultaneous equations \[ 2x\minus{}3y\equal{}8\] \[ 6y\minus{}4x\equal{}9\] $\textbf{(A)}\ x=4,y=0 \qquad \textbf{(B)}\ x=0,y=\dfrac{3}{2}\qquad \textbf{(C)}\ x=0,y=0 \qquad\\ \textbf{(D)}\ \text{There is no solution} \qquad \textbf{(E)}\ \text{There are an infinite number of solutions}$

Find the number of integers $n$ such that \[1+\left\lfloor\dfrac{100n}{101}\right\rfloor=\left\lceil\dfrac{99n}{100}\right\rceil.\]

A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance $d$ traveled by the two animals over time $t$ from start to finish?$\phantom{h}$ [asy] unitsize(0.4 cm); pair transx, transy; int i, j; int x, y; transx = (13,0); transy = (0,-9); for (i = 0; i <= 2; ++i) { for (j = 0; j <= 1; ++j) { if (i <= 1 || j <= 0) { for (x = 1; x <= 10; ++x) { draw(shift(i*transx + j*transy)*((x,0)--(x,5)),gray(0.7) + dashed); } for (y = 1; y <= 5; ++y) { draw(shift(i*transx + j*transy)*((0,y)--(10,y)),gray(0.7) + dashed); } draw(shift(i*transx + j*transy)*((0,0)--(11,0)),Arrow(6)); draw(shift(i*transx + j*transy)*((0,0)--(0,6)),Arrow(6)); label("time", (5,-0.5) + i*transx + j*transy); label(rotate(90)*"distance", (-0.5,2.5) + i*transx + j*transy); } }} draw((0,0)--(1.5,2.5)--(7.5,2.5)--(9,5),linewidth(1.5*bp)); draw((0,0)--(10,5),linewidth(1.5*bp)); draw(shift(transx)*((0,0)--(2.5,2.5)--(7.5,2.5)--(10,5)),linewidth(1.5*bp)); draw(shift(transx)*((0,0)--(9,5)),linewidth(1.5*bp)); draw(shift(2*transx)*((0,0)--(2.5,3)--(7,2)--(10,5)),linewidth(1.5*bp)); draw(shift(2*transx)*((0,0)--(9,5)),linewidth(1.5*bp)); draw(shift(transy)*((0,0)--(2.5,2.5)--(6.5,2.5)--(9,5)),linewidth(1.5*bp)); draw(shift(transy)*((0,0)--(7.5,2)--(10,5)),linewidth(1.5*bp)); draw(shift(transx + transy)*((0,0)--(2.5,2)--(7.5,3)--(10,5)),linewidth(1.5*bp)); draw(shift(transx + transy)*((0,0)--(9,5)),linewidth(1.5*bp)); label("(A)", (-1,6)); label("(B)", (-1,6) + transx); label("(C)", (-1,6) + 2*transx); label("(D)", (-1,6) + transy); label("(E)", (-1,6) + transx + transy); [/asy]

A circle of radius 1 is randomly placed in a 15-by-36 rectangle $ABCD$ so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal $AC$ is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

The points that satisfy the system $ x \plus{} y \equal{} 1,\, x^2 \plus{} y^2 < 25,$ constitute the following set: $ \textbf{(A)}\ \text{only two points} \qquad \\ \textbf{(B)}\ \text{an arc of a circle}\qquad \\ \textbf{(C)}\ \text{a straight line segment not including the end\minus{}points}\qquad \\ \textbf{(D)}\ \text{a straight line segment including the end\minus{}points}\qquad \\ \textbf{(E)}\ \text{a single point}$

A set of lattice points is called [i]good[/i] if it does not contain two points that form a line with slope $-1$ or slope $1$. Let $S = \{(x, y)\ |\ x, y \in \mathbb{Z}, 1 \le x, y \le 4\}$. Compute the number of non-empty good subsets of $S$. [i]Proposed by Lewis 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$

It is desired to construct a right triangle in the coordinate plane so that its legs are parallel to the $x$ and $y$ axes and so that the medians to the midpoints of the legs lie on the lines $y = 3x + 1$ and $y = mx + 2$. The number of different constants $m$ for which such a triangle exists is $ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \text{more than 3} $

Find the largest value for $\frac{y}{x}$ for pairs of real numbers $(x,y)$ which satisfy \[(x-3)^2 + (y-3)^2 = 6.\] $\textbf{(A) }3 + 2 \sqrt 2\qquad \textbf{(B) } 2 + \sqrt 3\qquad \textbf{(C ) }3 \sqrt 3\qquad \textbf{(D) }6\qquad \textbf{(E) }6 + 2 \sqrt 3$

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? $\textbf{(A)}\ \frac16 \qquad\textbf{(B)}\ \frac23 \qquad\textbf{(C)}\ \frac32 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 6$

On the coordinate plane, is there a line family of infinitely many lines $l_1,l_2,\cdots,l_n,\cdots$, satisfying the following? (1) Point$(1,1)\in l_n$ for all $n\in \mathbb{Z}_{+}$. (2) For all $n\in \mathbb{Z}_{+}$,$k_{n+1}=a_n-b_n$, where $k_{n+1}$ is the slope of $l_{n+1}$, $a_n,b_n$ are intercepts of $l_n$ on $x$-axis, $y$-axis. (3) $k_nk_{n+1}\geq0$ for all $n\in \mathbb{Z}_{+}$.

Let $P$ be a point in a square whose side are mirror. A ray of light comes from $P$ and with slope $\alpha$. We know that this ray of light never arrives to a vertex. We make an infinite sequence of $0,1$. After each contact of light ray with a horizontal side, we put $0$, and after each contact with a vertical side, we put $1$. For each $n\geq 1$, let $B_{n}$ be set of all blocks of length $n$, in this sequence. a) Prove that $B_{n}$ does not depend on location of $P$. b) Prove that if $\frac{\alpha}{\pi}$ is irrational, then $|B_{n}|=n+1$.