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 $ABC$.

The curve $y=y(x)$ satisfies $y'(0)=1.$ It satisfies the differential equation $(x^2 +9)y'' +(x^2 +4)y=0.$ Show that it crosses the $x$-axis between $$x= \frac{3}{2} \pi \;\;\; \text{and} \;\;\; x= \sqrt{\frac{63}{53}} \pi.$$

The line segment formed by $A(1, 2)$ and $B(3, 3)$ is rotated to the line segment formed by $A'(3, 1)$ and $B'(4, 3)$ about the point $P(r, s)$. What is $|r-s|$? $\text{A) } \frac{1}{4} \qquad \text{B) } \frac{1}{2} \qquad \text{C) } \frac{3}{4} \qquad \text{D) } \frac{2}{3} \qquad \text{E) } 1$

Given two points $P,\ Q$ on the parabola $C: y=x^2-x-2$ in the $xy$ plane. Note that the $x$ coodinate of $P$ is less than that of $Q$. (a) If the origin $O$ is the midpoint of the lines　egment $PQ$, then find the equation of the line $PQ$. (b) If the origin $O$ divides internally the line segment $PQ$ by 2:1, then find the equation of $PQ$. (c) If the origin $O$ divides internally the line segment $PQ$ by 2:1, find the area of the figure bounded by the parabola $C$ and the line $PQ$.

For a point $P = (a,a^2)$ in the coordinate plane, let $l(P)$ denote the line passing through $P$ with slope $2a$. Consider the set of triangles with vertices of the form $P_1 = (a_1, a_1^2), P_2 = (a_2, a_2^2), P_3 = (a_3, a_3^2)$, such that the intersection of the lines $l(P_1), l(P_2), l(P_3)$ form an equilateral triangle $\triangle$. Find the locus of the center of $\triangle$ as $P_1P_2P_3$ ranges over all such triangles.

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 $OA$.

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 $p+q+r$.

Cities $A$, $B$, $C$, $D$, and $E$ are connected by roads $\widetilde{AB}$, $\widetilde{AD}$, $\widetilde{AE}$, $\widetilde{BC}$, $\widetilde{BD}$, $\widetilde{CD}$, $\widetilde{DE}$. How many different routes are there from $A$ to $B$ that use each road exactly once? (Such a route will necessarily visit cities more than once.) [asy]unitsize(10mm); defaultpen(linewidth(1.2pt)+fontsize(10pt)); dotfactor=4; pair A=(1,0), B=(4.24,0), C=(5.24,3.08), D=(2.62,4.98), E=(0,3.08); dot (A); dot (B); dot (C); dot (D); dot (E); label("$A$",A,S); label("$B$",B,SE); label("$C$",C,E); label("$D$",D,N); label("$E$",E,W); guide squiggly(path g, real stepsize, real slope=45) { real len = arclength(g); real step = len / round(len / stepsize); guide squig; for (real u = 0; u < len; u += step){ real a = arctime(g, u); real b = arctime(g, u + step / 2); pair p = point(g, a); pair q = point(g, b); pair np = unit( rotate(slope) * dir(g,a)); pair nq = unit( rotate(0 - slope) * dir(g,b)); squig = squig .. p{np} .. q{nq}; } squig = squig .. point(g, length(g)){unit(rotate(slope)*dir(g,length(g)))}; return squig; } pen pp = defaultpen + 2.718; draw(squiggly(A--B, 4.04, 30), pp); draw(squiggly(A--D, 7.777, 20), pp); draw(squiggly(A--E, 5.050, 15), pp); draw(squiggly(B--C, 5.050, 15), pp); draw(squiggly(B--D, 4.04, 20), pp); draw(squiggly(C--D, 2.718, 20), pp); draw(squiggly(D--E, 2.718, -60), pp); [/asy] $ \textbf{(A)}\ 7 \qquad \textbf{(B)}\ 9\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 18 $

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 $

For $135^\circ < x < 180^\circ$, points $P=(\cos x, \cos^2 x), Q=(\cot x, \cot^2 x), R=(\sin x, \sin^2 x)$ and $S =(\tan x, \tan^2 x)$ are the vertices of a trapezoid. What is $\sin(2x)$? $ \textbf{(A)}\ 2-2\sqrt{2}\qquad\textbf{(B)}\ 3\sqrt{3}-6\qquad\textbf{(C)}\ 3\sqrt{2}-5\qquad\textbf{(D)}\ -\frac{3}{4}\qquad\textbf{(E)}\ 1-\sqrt{3} $

Let $a,\ b$ be positive real numbers. Consider the circle $C_1: (x-a)^2+y^2=a^2$ and the ellipse $C_2: x^2+\frac{y^2}{b^2}=1.$ (1) Find the condition for which $C_1$ is inscribed in $C_2$. (2) Suppose $b=\frac{1}{\sqrt{3}}$ and $C_1$ is inscribed in $C_2$. Find the coordinate $(p,\ q)$ of the point of tangency in the first quadrant for $C_1$ and $C_2$. (3) Under the condition in (1), find the area of the part enclosed by $C_1,\ C_2$ for $x\geq p$. 60 point

Let $f(x)$ be a continuous function, whose first and second derivatives are continuous on $[0,2\pi]$ and $f''(x) \geq 0$ for all $x$ in $[0,2\pi]$. Show that \[\int_{0}^{2\pi} f(x)\cos x dx \geq 0\]

Let $ a<a'<b<b'$ be real numbers and let the real function $ f$ be continuous on the interval $ [a,b']$ and differentiable in its interior. Prove that there exist $ c \in (a,b), c'\in (a',b')$ such that \[ f(b)\minus{}f(a)\equal{}f'(c)(b\minus{}a),\] \[ f(b')\minus{}f(a')\equal{}f'(c')(b'\minus{}a'),\] and $ c<c'$. [i]B. Szokefalvi Nagy[/i]

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 $\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }\sqrt{2}\qquad \textbf{(E) }2\sqrt{2}$

Square $PQRS$ lies in the first quadrant. Points $(3,0), (5,0), (7,0),$ and $(13,0)$ lie on lines $SP, RQ, PQ$, and $SR$, respectively. What is the sum of the coordinates of the center of the square $PQRS$? $ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 6.2\qquad\textbf{(C)}\ 6.4\qquad\textbf{(D)}\ 6.6\qquad\textbf{(E)}\ 6.8 $

Let $l,\ m$ be the tangent lines passing through the point $A(a,\ a-1)$ on the line $y=x-1$ and touch the parabola $y=x^2$. Note that the slope of $l$ is greater than that of $m$. (1) Exress the slope of $l$ in terms of $a$. (2) Denote $P,\ Q$ be the points of tangency of the lines $l,\ m$ and the parabola $y=x^2$. Find the minimum area of the part bounded by the line segment $PQ$ and the parabola $y=x^2$. (3) Find the minimum distance between the parabola $y=x^2$ and the line $y=x-1$.

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 $m + n$.

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

The solution of the equations \begin{align*} 2x-3y&=7 \\ 4x-6y &=20 \\ \end{align*} is: $ \textbf{(A)}\ x=18, y=12 \qquad \textbf{(B)}\ x=0, y=0 \qquad \textbf{(C)}\ \text{There is no solution} \\ \textbf{(D)}\ \text{There are an unlimited number of solutions} \qquad \textbf{(E)}\ x=8, y=5$

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.)

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.