The set of all points $P$ such that the sum of the (undirected) distances from $P$ to two fixed points $A$ and $B$ equals the distance between $A$ and $B$ is $\textbf{(A) }\text{the line segment from }A\text{ to }B\qquad$ $\textbf{(B) }\text{the line passing through }A\text{ and }B\qquad$ $\textbf{(C) }\text{the perpendicular bisector of the line segment from }A\text{ to }B\qquad$ $\textbf{(D) }\text{an elllipse having positive area}\qquad$ $\textbf{(E) }\text{a parabola}$

$A(0,2)$, and two points $B,C$ on parabola $y^2=x+4$ satisfy that $AB\perp BC$. Find the range value of $y_C$.

Suppose $w,x,y,z$ satisfy \begin{align*}w+x+y+z&=25,\\wx+wy+wz+xy+xz+yz&=2y+2z+193\end{align*} The largest possible value of $w$ can be expressed in lowest terms as $w_1/w_2$ for some integers $w_1,w_2>0$. Find $w_1+w_2$.

A parabola with equation $ y \equal{} ax^2 \plus{} bx \plus{} c$ is reflected about the $ x$-axis. The parabola and its reflection are translated horizontally five units in opposite directions to become the graphs of $ y \equal{} f(x)$ and $ y \equal{} g(x)$, respectively. Which of the following describes the graph of $ y \equal{} (f \plus{} g)(x)$? $ \textbf{(A)}\ \text{a parabola tangent to the }x\text{ \minus{} axis}$ $ \textbf{(B)}\ \text{a parabola not tangent to the }x\text{ \minus{} axis} \qquad \textbf{(C)}\ \text{a horizontal line}$ $ \textbf{(D)}\ \text{a non \minus{} horizontal line} \qquad \textbf{(E)}\ \text{the graph of a cubic function}$

The line $L$ is parallel to the plane $y=z$ and meets the parabola $2x=y^2 ,z=0$ and the parabola $3x=z^2, y=0$. Prove that if $L$ moves freely subject to these constraints then it generates the surface $x=(y-z)\left(\frac{y}{2}-\frac{z}{3}\right)$.

A standard parabola is the graph of a quadratic polynomial $y = x^2 + ax + b$ with leading co\"efficient 1. Three standard parabolas with vertices $V1, V2, V3$ intersect pairwise at points $A1, A2, A3$. Let $A \mapsto s(A)$ be the reflection of the plane with respect to the $x$-axis. Prove that standard parabolas with vertices $s (A1), s (A2), s (A3)$ intersect pairwise at the points $s (V1), s (V2), s (V3)$.

A parabola is drawn on the coordinate plane - the graph of a square trinomial. The vertices of triangle $ABC$ lie on this parabola so that the bisector of angle $\angle BAC$ is parallel to the axis $Ox$ . Prove that the midpoint of the median drawn from vertex $A$ lies on the axis of the parabola.

Let $ABC$ be an acute, scalene triangle, and let $P$ be an arbitrary point in its interior. Let $\mathcal{P}_A$ be the parabola with focus $P$ and directrix $BC$, and define $\mathcal{P}_B$ and $\mathcal{P}_C$ similarly. (a) Show that if $Q$ is an intersection point of $\mathcal{P}_B$ and $\mathcal{P}_C$, then $P$ and $Q$ are on the same side of $AB$, and $P$ and $Q$ are on the same side of $AC$. (b) You are given that $\mathcal{P}_B$ and $\mathcal{P}_C$ intersect at exactly two points. Let $\ell_A$ be the line between these points, and define $\ell_B$ and $\ell_C$ similarly. Show that $\ell_A$, $\ell_B$, and $\ell_C$ concur. [i]Note: A parabola with focus point $X$ and directrix line $\ell$ is the set of all points $Z$ that are the same distance from $X$ and $\ell$.[/i]

Let $l$ be the tangent line at the point $(t,\ t^2)\ (0<t<1)$ on the parabola $C: y=x^2$. Denote by $S_1$ the area of the part enclosed by $C,\ l$ and the $x$-axis, denote by $S_2$ of the area of the part enclosed by $C,\ l$ and the line $x=1$. Find the minimum value of $S_1+S_2$.

A parabola has focus $F$ and vertex $V$ , where $VF = 1$0. Let $AB$ be a chord of length $100$ that passes through $F$. Determine the area of $\vartriangle VAB$.

A line through the origin passes through the curve whose equation is $5y=2x^2-9x+10$ at two points whose $x-$coordinates add up to $77.$ Find the slope of the line.

Let $ a\geq 2$ be a real number; with the roots $ x_{1}$ and $ x_{2}$ of the equation $ x^2\minus{}ax\plus{}1\equal{}0$ we build the sequence with $ S_{n}\equal{}x_{1}^n \plus{} x_{2}^n$. [b]a)[/b]Prove that the sequence $ \frac{S_{n}}{S_{n\plus{}1}}$, where $ n$ takes value from $ 1$ up to infinity, is strictly non increasing. [b]b)[/b]Find all value of $ a$ for the which this inequality hold for all natural values of $ n$ $ \frac{S_{1}}{S_{2}}\plus{}\cdots \plus{}\frac{S_{n}}{S_{n\plus{}1}}>n\minus{}1$

Let $k$ be a positive real number. In the $X-Y$ coordinate plane, let $S$ be the set of all points of the form $(x,x^2+k)$ where $x\in\mathbb{R}$. Let $C$ be the set of all circles whose center lies in $S$, and which are tangent to $X$-axis. Find the minimum value of $k$ such that any two circles in $C$ have at least one point of intersection.

In the coordinate plane, suppose that the parabola $C: y=-\frac{p}{2}x^2+q\ (p>0,\ q>0)$ touches the circle with radius 1 centered on the origin at distinct two points. Find the minimum area of the figure enclosed by the part of $y\geq 0$ of $C$ and the $x$-axis.

All vertices of triangles $ABC$ and $A_1B_1C_1$ lie on the hyperbola $y=1/x$. It is known that $AB \parallel A_1B_1$ and $BC \parallel B_1C_1$. Prove that $AC_1 \parallel A_1C$. (I. Gorodnin)

If $0<a<b$, given two fixed points $A(a,0),B(b,0)$. Draw lines $l$ passes $A$, $m$ passes $B$. They have four different intersections with parabola $y^2=x$. If the four points are concyclic, find the path of $P(P=l\cap m)$.

1.Let $ x \equal{} \alpha ,\ \beta \ (\alpha < \beta )$ are $ x$ coordinates of the intersection points of a parabola $ y \equal{} ax^2 \plus{} bx \plus{} c\ (a\neq 0)$ and the line $ y \equal{} ux \plus{} v$. Prove that the area of the region bounded by these graphs is $ \boxed{\frac {|a|}{6}(\beta \minus{} \alpha )^3}$. 2. Let $ x \equal{} \alpha ,\ \beta \ (\alpha < \beta )$ are $ x$ coordinates of the intersection points of parabolas $ y \equal{} ax^2 \plus{} bx \plus{} c$ and $ y \equal{} px^2 \plus{} qx \plus{} r\ (ap\neq 0)$. Prove that the area of the region bounded by these graphs is $ \boxed{\frac {|a \minus{} p|}{6}(\beta \minus{} \alpha )^3}$.

The given parabola $y=ax^2+bx+c$ doesn't intersect the $X$-axis and passes from the points $A(-2,1)$ and $B(2,9)$. Find all the possible values of the $x$ coordinates of the vertex of this parabola.

The parabola $y = x^2$ is drawn in the coordinate plane and then the axes are erased so that the whole parabola stays on the picture but the origin is not shown on it. Reconstruct the axes with compass and ruler alone. (A Egorov)

Let $a, b, c$ be real numbers such that $a \neq 0$. Consider the parabola with equation \[ y = ax^2 + bx + c, \] and the lines defined by the six equations \begin{align*} &y = ax + b, \quad & y = bx + c, \qquad \quad & y = cx + a, \\ &y = bx + a, \quad & y = cx + b, \qquad \quad & y = ax + c. \end{align*} Suppose that the parabola intersects each of these lines in at most one point. Determine the maximum and minimum possible values of $\frac{c}{a}$.

Various points $x_1,..., x_n$ ($n \ge 3$) are randomly located on the $Ox$ axis. Construct all parabolas defined by the monic square trinomials and intersecting the Ox axis at these points (and not intersecting axis at other points). Let$ y = f_1$, $...$ , $y = f_m$ are functions that define these parabolas. Prove that the parabola $y = f_1 +...+ f_m$ intersects the $Ox$ axis at two points.

Given a triangle $ABC$. A point $X$ is chosen on a side $AC$. Some circle passes through $X$, touches the side $AC$ and intersects the circumcircle of triangle $ABC$ in points $M$ and $N$ such that the segment $MN$ bisects $BX$ and intersects sides $AB$ and $BC$ in points $P$ and $Q$. Prove that the circumcircle of triangle $PBQ$ passes through a fixed point different from $B$. [b]proposed by Sergej Berlov[/b]

A circle centered at a point $(0, 1)$ on the coordinate plane intersects the parabola $y = x^2$ at four points: $A, B, C, D.$ Find the largest possible value of the area of ​​the quadrilateral $ABCD$.

$A,B,C$ 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.

For some positive integers $p$, there is a quadrilateral $ABCD$ with positive integer side lengths, perimeter $p$, right angles at $B$ and $C$, $AB=2$, and $CD=AD$. How many different values of $p<2015$ are possible? $\textbf{(A) }30\qquad\textbf{(B) }31\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$