What is the degree measure of the acute angle formed by lines with slopes $2$ and $\tfrac{1}{3}$? $\textbf{(A)}~30\qquad\textbf{(B)}~37.5\qquad\textbf{(C)}~45\qquad\textbf{(D)}~52.5\qquad\textbf{(E)}~60$

For every pair of reals $0 < a < b < 1$, we define sequences $\{x_n\}_{n \ge 0}$ and $\{y_n\}_{n \ge 0}$ by $x_0 = 0$, $y_0 = 1$, and for each integer $n \ge 1$: \begin{align*} x_n & = (1 - a) x_{n - 1} + a y_{n - 1}, \\ y_n & = (1 - b) x_{n - 1} + b y_{n - 1}. \end{align*} The [i]supermean[/i] of $a$ and $b$ is the limit of $\{x_n\}$ as $n$ approaches infinity. Over all pairs of real numbers $(p, q)$ satisfying $\left (p - \textstyle\frac{1}{2} \right)^2 + \left (q - \textstyle\frac{1}{2} \right)^2 \le \left(\textstyle\frac{1}{10}\right)^2$, the minimum possible value of the supermean of $p$ and $q$ can be expressed as $\textstyle\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m + n$. [i]Proposed by Lewis Chen[/i]

A line $\ell$ with slope of $\frac{1}{3}$ insects the ellipse $C:\frac{x^2}{36}+\frac{y^2}{4}=1$ at points $A,B$ and the point $P\left( 3\sqrt{2} , \sqrt{2}\right)$ is above the line $\ell$. [list] [b](1)[/b] Prove that the locus of the incenter of triangle $PAB$ is a segment, [b](2)[/b] If $\angle APB=\frac{\pi}{3}$, then find the area of triangle $PAB$.[/list]

Let $M$ and $N$ be points on the sides $BC$ and $CD$, respectively, of a square $ABCD$. If $|BM|=21$, $|DN|=4$, and $|NC|=24$, what is $m(\widehat{MAN})$? $ \textbf{(A)}\ 15^\circ \qquad\textbf{(B)}\ 30^\circ \qquad\textbf{(C)}\ 37^\circ \qquad\textbf{(D)}\ 45^\circ \qquad\textbf{(E)}\ 60^\circ $

$A,B,C$ and $D$ are points on the parabola $y = x^2$ such that $AB$ and $CD$ intersect on the $y$-axis. Determine the $x$-coordinate of $D$ in terms of the $x$-coordinates of $A,B$ and $C$, which are $a, b$ and $c$ respectively.

An equilateral triangle is inscribed in the ellipse whose equation is $x^2+4y^2=4.$ One vertex of the triangle is $(0,1),$ one altitude is contained in the $y$-axis, and the length of each side is $\sqrt{\frac mn},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

This problem is easy but nobody solved it. point $A$ moves in a line with speed $v$ and $B$ moves also with speed $v'$ that at every time the direction of move of $B$ goes from $A$.We know $v \geq v'$.If we know the point of beginning of path of $A$, then $B$ must be where at first that $B$ can catch $A$.

Where is AMC 10a No.19? Thanks

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 square $ABCD$ (labeled clockwise), let $P$ be any point on $BC$ and construct square $APRS$ (labeled clockwise). Prove that line $CR$ is tangent to the circumcircle of triangle $ABC$.

A ray is drawn from the origin tangent to the graph of the upper part of the hyperbola $y^2=x^2-x+1$ in the first quadrant. This ray makes an angle of $\theta$ with the positive $x$-axis. Compute $\cos\theta$.

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$.

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$

Given that $x^2 + y^2 = 14x + 6y + 6$, what is the largest possible value that $3x + 4y$ can have? $\text{(A)}\ 72 \qquad \text{(B)}\ 73 \qquad \text{(C)}\ 74 \qquad \text{(D)}\ 75\qquad \text{(E)}\ 76$

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?

Three circles, each of radius 3, are drawn with centers at $(14,92)$, $(17,76)$, and $(19,84)$. A line passing through $(17,76)$ is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the absolute value of the slope of this line?

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$

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 $a+b+c=2014$.

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 $A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8$ be a regular octagon. Let $M_1$, $M_3$, $M_5$, and $M_7$ be the midpoints of sides $\overline{A_1 A_2}$, $\overline{A_3 A_4}$, $\overline{A_5 A_6}$, and $\overline{A_7 A_8}$, respectively. For $i = 1, 3, 5, 7$, ray $R_i$ is constructed from $M_i$ towards the interior of the octagon such that $R_1 \perp R_3$, $R_3 \perp R_5$, $R_5 \perp R_7$, and $R_7 \perp R_1$. Pairs of rays $R_1$ and $R_3$, $R_3$ and $R_5$, $R_5$ and $R_7$, and $R_7$ and $R_1$ meet at $B_1$, $B_3$, $B_5$, $B_7$ respectively. If $B_1 B_3 = A_1 A_2$, then $\cos 2 \angle A_3 M_3 B_1$ can be written in the form $m - \sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.

Triangle $ABC$ lies entirely in the first quadrant of the Cartesian plane, and its sides have slopes $63$, $73$, $97$. Suppose the curve $\mathcal V$ with equation $y=(x+3)(x^2+3)$ passes through the vertices of $ABC$. Find the sum of the slopes of the three tangents to $\mathcal V$ at each of $A$, $B$, $C$. [i]Proposed by Akshaj[/i]

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}$

Line $\ell_1$ has equation $3x-2y=1$ and goes through $A=(-1,-2)$. Line $\ell_2$ has equation $y=1$ and meets line $\ell_1$ at point $B$. Line $\ell_3$ has positive slope, goes through point $A$, and meets $\ell_2$ at point $C$. The area of $\triangle ABC$ is $3$. What is the slope of $\ell_3$? $ \textbf{(A)}\ \frac{2}{3}\qquad\textbf{(B)}\ \frac{3}{4}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{4}{3}\qquad\textbf{(E)}\ \frac{3}{2} $

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

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 $f(2)$?