Consider the $ 12$-sided polygon $ ABCDEFGHIJKL$, as shown. Each of its sides has length $ 4$, and each two consecutive sides form a right angle. Suppose that $ \overline{AG}$ and $ \overline{CH}$ meet at $ M$. What is the area of quadrilateral $ ABCM$? [asy]unitsize(13mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(1,3), B=(2,3), C=(2,2), D=(3,2), Ep=(3,1), F=(2,1), G=(2,0), H=(1,0), I=(1,1), J=(0,1), K=(0,2), L=(1,2); pair M=intersectionpoints(A--G,H--C)[0]; draw(A--B--C--D--Ep--F--G--H--I--J--K--L--cycle); draw(A--G); draw(H--C); dot(M); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,NE); label("$D$",D,NE); label("$E$",Ep,SE); label("$F$",F,SE); label("$G$",G,SE); label("$H$",H,SW); label("$I$",I,SW); label("$J$",J,SW); label("$K$",K,NW); label("$L$",L,NW); label("$M$",M,W);[/asy]$ \textbf{(A)}\ \frac {44}{3}\qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ \frac {88}{5}\qquad \textbf{(D)}\ 20 \qquad \textbf{(E)}\ \frac {62}{3}$

A square in the coordinate plane has vertices whose $y$-coordinates are $0$, $1$, $4$, and $5$. What is the area of the square? $ \textbf{(A)}\ 16\qquad\textbf{(B)}\ 17\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 26\qquad\textbf{(E)}\ 27 $

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 $m + n$.

Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of $\triangle ABC$? [asy] size(200); defaultpen(linewidth(.6pt)+fontsize(12pt)); dotfactor=4; draw((0,0)--(0,2)); draw((0,0)--(1,0)); draw((1,0)--(1,2)); draw((0,1)--(2,1)); draw((0,0)--(1,2)); draw((0,2)--(2,1)); draw((0,2)--(2,2)); draw((2,1)--(2,2)); label("$A$",(0,2),NW); label("$B$",(1,2),N); label("$C$",(4/5,1.55),W); dot((0,2)); dot((1,2)); dot((4/5,1.6)); dot((2,1)); dot((0,0)); [/asy] $ \textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{5}\qquad\textbf{(C)}\ \frac{2}{9}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{\sqrt2}{4} $

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

Let $ f(x)\equal{}x^2\plus{}3$ and $ y\equal{}g(x)$ be the equation of the line with the slope $ a$, which pass through the point $ (0,\ f(0))$ . Find the maximum and minimum values of $ I(a)\equal{}3\int_{\minus{}1}^1 |f(x)\minus{}g(x)|\ dx$.

A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labelled $A$. The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $\tfrac{r-\sqrt{s}}{t}$, where $r$, $s$, and $t$ are positive integers, and $r+s+t<1000$. Find $r+s+t$.

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]

Three points $A\left(0,\frac{4}{3}\right),B(-1,0),C(1,0)$ are given. The distance from $P$ to line $BC$ is the geometric mean of that from $P$ to lines $AB$ and $AC$. [b](a)[/b] Find the path equation of point $P$. [b](b)[/b] If line $L$ passes $D$ ($D$ is the incenter of $\triangle ABC$ ), and it has three common points with the path of $P$, find the range value of slope $k$ of line $L$.

A line with slope $ 3$ intersects a line with slope $ 5$ at the point $ (10, 15)$. What is the distance between the $ x$-intercepts of these two lines? $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 20$

Take a point $P\left(\frac 12,\ \frac 14\right)$ on the coordinate plane. Let two points $Q(\alpha ,\ \alpha ^ 2),\ R(\beta ,\ \beta ^2)$ move in such a way that 3 points $P,\ Q,\ R$ form an isosceles triangle with the base $QR$, find the locus of the barycenter $G(X,\ Y)$ of $\triangle{PQR}$. [i]2011 Tokyo University entrance exam[/i]

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} $

Complex numbers $a$, $b$ and $c$ are the zeros of a polynomial $P(z) = z^3+qz+r$, and $|a|^2+|b|^2+|c|^2=250$. The points corresponding to $a$, $b$, and $c$ in the complex plane are the vertices of a right triangle with hypotenuse $h$. Find $h^2$.

If $m > 0$ and the points $(m,3)$ and $(1,m)$ lie on a line with slope $m$, then $m = $ $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ \sqrt{2}\qquad\textbf{(C)}\ \sqrt{3}\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ \sqrt{5} $

Where is AMC 10a No.19? Thanks

Let $m$ and $n$ be positive integers. A sequence of points $(A_0,A_1,\ldots,A_n)$ on the Cartesian plane is called [i]interesting[/i] if $A_i$ are all lattice points, the slopes of $OA_0,OA_1,\cdots,OA_n$ are strictly increasing ($O$ is the origin) and the area of triangle $OA_iA_{i+1}$ is equal to $\frac{1}{2}$ for $i=0,1,\ldots,n-1$. Let $(B_0,B_1,\cdots,B_n)$ be a sequence of points. We may insert a point $B$ between $B_i$ and $B_{i+1}$ if $\overrightarrow{OB}=\overrightarrow{OB_i}+\overrightarrow{OB_{i+1}}$, and the resulting sequence $(B_0,B_1,\ldots,B_i,B,B_{i+1},\ldots,B_n)$ is called an [i]extension[/i] of the original sequence. Given two [i]interesting[/i] sequences $(C_0,C_1,\ldots,C_n)$ and $(D_0,D_1,\ldots,D_m)$, prove that if $C_0=D_0$ and $C_n=D_m$, then we may perform finitely many [i]extensions[/i] on each sequence until the resulting two sequences become identical.

Prove that the number of incidences of $n$ distinct points on $n$ distinct lines in plane is $\mathcal O (n^{\frac{4}{3}})$. Find a configuration for which $\Omega (n^{\frac{4}{3}})$ incidences happens.

Let triangle $ABC$ be a right triangle in the xy-plane with a right angle at $C$. Given that the length of the hypotenuse $AB$ is 60, and that the medians through $A$ and $B$ lie along the lines $y=x+3$ and $y=2x+4$ respectively, find the area of triangle $ABC$.

Let $w$ and $z$ be complex numbers such that $|w| = 1$ and $|z| = 10$. Let $\theta = \arg\left(\tfrac{w-z}{z}\right)$. The maximum possible value of $\tan^2 \theta$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. (Note that $\arg(w)$, for $w \neq 0$, denotes the measure of the angle that the ray from $0$ to $w$ makes with the positive real axis in the complex plane.

A cubic polynomial $f$ with a positive leading coefficient has three different positive zeros. Show that $f'(a)+ f'(b)+ f'(c) > 0$.

$ 2n\plus{}1$ lines are drawn in the plane, in such a way that every 3 lines define a triangle with no right angles. What is the maximal possible number of acute triangles that can be made in this way?

Let $ABCD$ be a square, $E$ and $F$ midpoints of $AB$ and $AD$ respectively, and $P$ the intersection of $CF$ and $DE$. a) Show that $DE \perp CF$. b) Determine the ratio $CF : PC : EP$

A line passes through $ A(1,1)$ and $ B(100,1000)$. How many other points with integer coordinates are on the line and strictly between $ A$ and $ B$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 9$

The slope of the line $\frac{x}{3} + \frac{y}{2} = 1$ is $ \textbf{(A)}\ -\frac{3}{2}\qquad\textbf{(B)}\ -\frac{2}{3}\qquad\textbf{(C)}\ \frac{1}{3}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{3}{2} $

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]