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 $OA$.

Given the four equations: $\textbf{(1)}\ 3y-2x=12 \qquad \textbf{(2)}\ -2x-3y=10 \qquad \textbf{(3)}\ 3y+2x=12 \qquad \textbf{(4)}\ 2y+3x=10$ The pair representing the perpendicular lines is: $\textbf{(A)}\ \text{(1) and (4)} \qquad \textbf{(B)}\ \text{(1) and (3)} \qquad \textbf{(C)}\ \text{(1) and (2)} \qquad \textbf{(D)}\ \text{(2) and (4)} \qquad \textbf{(E)}\ \text{(2) and (3)}$

Square $\mathcal S$ has vertices $(1,0)$, $(0,1)$, $(-1,0)$ and $(0,-1)$. Points $P$ and $Q$ are independently selected, uniformly at random, from the perimeter of $\mathcal S$. Determine, with proof, the probability that the slope of line $PQ$ is positive. [i]Proposed by Isabella Grabski[/i]

Let $n,k$ be integers such that $n\geq k\geq3$. Consider $n+1$ points in a plane (there is no three collinear points) and $k$ different colors, then, we color all the segments that connect every two points. We say that an angle is good if its vertex is one of the initial set, and its two sides aren't the same color. Show that there exist a coloration such that the \\ total number of good angles is greater than $n \binom{k}{2} \lfloor(\frac{n}{k})\rfloor^2$

A sled loaded with children starts from rest and slides down a snowy $25^\circ$ (with respect to the horizontal) incline traveling $85$ meters in $17$ seconds. Ignore air resistance. What is the coefficient of kinetic friction between the sled and the slope? $ \textbf {(A) } 0.36 \qquad \textbf {(B) } 0.40 \qquad \textbf {(C) } 0.43 \qquad \textbf {(D) } 1.00 \qquad \textbf {(E) } 2.01 $

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 \[\int_{0}^{2\pi} f(x)\cos x dx \geq 0\]

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]

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 $PQ$.

Find all values of $ \alpha$ for which the curves $ y\equal{}\alpha x^2\plus{}\alpha x\plus{}\frac1{24}$ and $ x\equal{}\alpha y^2\plus{}\alpha y\plus{}\frac1{24}$ are tangent to each other.

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 rectangle formed by the lines of checkered paper is divided into figures of three kinds: isosceles right triangles (1) with base of two units, squares (2) with unit side, and parallelograms (3) formed by two sides and two diagonals of unit squares (figures may be oriented in any way). Prove that the number of figures of the third kind is even. [img]http://up.iranblog.com/Files7/dda310bab8b6455f90ce.jpg[/img]

Let $ n$ be a positive integer. Consider a rectangle $ (90n\plus{}1)\times(90n\plus{}5)$ consisting of unit squares. Let $ S$ be the set of the vertices of these squares. Prove that the number of distinct lines passing through at least two points of $ S$ is divisible by $ 4$.

Square $ABCD$ has side length $68$. Let $E$ be the midpoint of segment $\overline{CD}$, and let $F$ be the point on segment $\overline{AB}$ a distance $17$ from point $A$. Point $G$ is on segment $\overline{EF}$ so that $\overline{EF}$ is perpendicular to segment $\overline{GD}$. The length of segment $\overline{BG}$ can be written as $m\sqrt{n}$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.

Given the line $y = \dfrac{3}{4}x + 6$ and a line $L$ parallel to the given line and $4$ units from it. A possible equation for $L$ is: $\textbf{(A)}\ y = \dfrac{3}{4}x + 1 \qquad \textbf{(B)}\ y = \dfrac{3}{4}x\qquad \textbf{(C)}\ y = \dfrac{3}{4}x -\dfrac{2}{3} \qquad$ $ \textbf{(D)}\ y = \dfrac{3}{4}x -1 \qquad \textbf{(E)}\ y = \dfrac{3}{4}x + 2$

In rectangle $ABCD$, $AB=1$, $BC=2$, and points $E$, $F$, and $G$ are midpoints of $\overline{BC}$, $\overline{CD}$, and $\overline{AD}$, respectively. Point $H$ is the midpoint of $\overline{GE}$. What is the area of the shaded region? [asy] import graph; size(9cm); pen dps = fontsize(10); defaultpen(dps); pair D = (0,0); pair F = (1/2,0); pair C = (1,0); pair G = (0,1); pair E = (1,1); pair A = (0,2); pair B = (1,2); pair H = (1/2,1); // do not look pair X = (1/3,2/3); pair Y = (2/3,2/3); draw(A--B--C--D--cycle); draw(G--E); draw(A--F--B); draw(D--H--C); filldraw(H--X--F--Y--cycle,grey); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); label("$E$",E,E); label("$F$",F,S); label("$G$",G,W); label("$H$",H,N); label("$\displaystyle\frac12$",(0.25,0),S); label("$\displaystyle\frac12$",(0.75,0),S); label("$1$",(1,0.5),E); label("$1$",(1,1.5),E); [/asy] $ \textbf{(A)}\ \dfrac1{12}\qquad\textbf{(B)}\ \dfrac{\sqrt3}{18}\qquad\textbf{(C)}\ \dfrac{\sqrt2}{12}\qquad\textbf{(D)}\ \dfrac{\sqrt3}{12}\qquad\textbf{(E)}\ \dfrac16 $

Let $ ABC$ be a triangle with $ AB \equal{} 5$, $ BC \equal{} 4$ and $ AC \equal{} 3$. Let $ \mathcal P$ and $ \mathcal Q$ be squares inside $ ABC$ with disjoint interiors such that they both have one side lying on $ AB$. Also, the two squares each have an edge lying on a common line perpendicular to $ AB$, and $ \mathcal P$ has one vertex on $ AC$ and $ \mathcal Q$ has one vertex on $ BC$. Determine the minimum value of the sum of the areas of the two squares. [asy]import olympiad; import math; import graph; unitsize(1.5cm); pair A, B, C; A = origin; B = A + 5 * right; C = (9/5, 12/5); pair X = .7 * A + .3 * B; pair Xa = X + dir(135); pair Xb = X + dir(45); pair Ya = extension(X, Xa, A, C); pair Yb = extension(X, Xb, B, C); pair Oa = (X + Ya)/2; pair Ob = (X + Yb)/2; pair Ya1 = (X.x, Ya.y); pair Ya2 = (Ya.x, X.y); pair Yb1 = (Yb.x, X.y); pair Yb2 = (X.x, Yb.y); draw(A--B--C--cycle); draw(Ya--Ya1--X--Ya2--cycle); draw(Yb--Yb1--X--Yb2--cycle); label("$A$", A, W); label("$B$", B, E); label("$C$", C, N); label("$\mathcal P$", Oa, origin); label("$\mathcal Q$", Ob, origin);[/asy]

Let $ABC$ be a triangle with $|AB|=|AC|$. $D$ is the midpoint of $[BC]$. $E$ is the foot of the altitude from $D$ to $AC$. $BE$ cuts the circumcircle of triangle $ABD$ at $B$ and $F$. $DE$ and $AF$ meet at $G$. Prove that $|DG|=|GE|$

If each of two intersecting lines intersects a hyperbola and neither line is tangent to the hyperbola, then the possible number of points of intersection with the hyperbola is: $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 2\text{ or }3 \qquad\textbf{(C)}\ 2\text{ or }4 \qquad\textbf{(D)}\ 3\text{ or }4 \qquad\textbf{(E)}\ 2,3,\text{ or }4$

If $x$ and $y$ are real numbers with $(x+y)^4=x-y$, what is the maximum possible value of $y$?

Let $w_{1}$ and $w_{2}$ denote the circles $x^{2}+y^{2}+10x-24y-87=0$ and $x^{2}+y^{2}-10x-24y+153=0$, respectively. Let $m$ be the smallest positive value of $a$ for which the line $y=ax$ contains the center of a circle that is externally tangent to $w_{2}$ and internally tangent to $w_{1}$. Given that $m^{2}=p/q$, where $p$ and $q$ are relatively prime integers, find $p+q$.

In the $xy$-plane, a circle of radius $4$ with center on the positive $x$-axis is tangent to the $y$-axis at the origin, and a circle with radius $10$ with center on the positive $y$-axis is tangent to the $x$-axis at the origin. What is the slope of the line passing through the two points at which these circles intersect? $\textbf{(A)}\ \dfrac{2}{7} \qquad\textbf{(B)}\ \dfrac{3}{7} \qquad\textbf{(C)}\ \dfrac{2}{\sqrt{29}} \qquad\textbf{(D)}\ \dfrac{1}{\sqrt{29}} \qquad\textbf{(E)}\ \dfrac{2}{5}$

A lattice point is a point in the plane with integer coordinates. How many lattice points are on the line segment whose endpoints are (3,17) and (48,281)? (Include both endpoints of the segment in your count.) $\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 46$

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]

Given a right triangular prism $A_1B_1C_1 - ABC$ with $\angle BAC = \frac{\pi}{2}$ and $AB = AC = AA_1$, let $G$, $E$ be the midpoints of $A_1B_1$, $CC_1$ respectively, and $D$, $F$ be variable points lying on segments $AC$, $AB$ (not including endpoints) respectively. If $GD \bot EF$, the range of the length of $DF$ is ${ \textbf{(A)}\ [\frac{1}{\sqrt{5}}, 1)\qquad\textbf{(B)}\ [\frac{1}{5}, 2)\qquad\textbf{(C)}\ [1, \sqrt{2})\qquad\textbf{(D)}} [\frac{1}{\sqrt{2}}, \sqrt{2})\qquad $

Given a scalene acute triangle $ ABC$ with $ AC>BC$ let $ F$ be the foot of the altitude from $ C$. Let $ P$ be a point on $ AB$, different from $ A$ so that $ AF\equal{}PF$. Let $ H,O,M$ be the orthocenter, circumcenter and midpoint of $ [AC]$. Let $ X$ be the intersection point of $ BC$ and $ HP$. Let $ Y$ be the intersection point of $ OM$ and $ FX$ and let $ OF$ intersect $ AC$ at $ Z$. Prove that $ F,M,Y,Z$ are concyclic.