Coordinate axes (without any marks, with the same scale) and the graph of a quadratic trinomial $y = x^2 + ax + b$ are drawn in the plane. The numbers $a$ and $b$ are not known. How to draw a unit segment using only ruler and compass?

The parabolas $y=ax^2-2$ and $y=4-bx^2$ intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area $12$. What is $a+b$? $\textbf{(A) }1\qquad\textbf{(B) }1.5\qquad\textbf{(C) }2\qquad\textbf{(D) }2.5\qquad\textbf{(E) }3$

For any positive integer $n$, $A_n$ and $B_n$ are intersection of parabola $y=(n^2+n)x^2-(2n+1)x+1$ and $x$-axis. Then, the value of $|A_1B_1|+|A_2B_2|+\cdots+|A_{1992}B_{1992}|$ is $\text{(A)}\frac{1991}{1992}\qquad\text{(B)}\frac{1992}{1993}\qquad\text{(C)}\frac{1991}{1993}\qquad\text{(D)}\frac{1993}{1992}$

For constants $a,\ b,\ c,\ d$ consider a process such that the point $(p,\ q)$ is mapped onto the point $(ap+bq,\ cp+dq)$. Note : $(a,\ b,\ c,\ d)\neq (1,\ 0,\ 0,\ 1)$. Let $k$ be non-zero constant. All points of the parabola $C: y=x^2-x+k$ are mapped onto $C$ by the process. (1) Find $a,\ b,\ c,\ d$. (2) Let $A'$ be the image of the point $A$ by the process. Find all values of $k$ and the coordinates of $A$ such that the tangent line of $C$ at $A$ and the tangent line of $C$ at $A'$ formed by the process are perpendicular at the origin.

Equations of non-intersecting curves are $y = ax^2 + bx + c$ and $y = dx^2 + ex + f$ where $ad < 0.$ Prove that there is a line of the plane which does not meet either of the curves.

$M$ is midpoint of $BC$.$P$ is an arbitary point on $BC$. $C_{1}$ is tangent to big circle.Suppose radius of $C_{1}$ is $r_{1}$ Radius of $C_{4}$ is equal to radius of $C_{1}$ and $C_{4}$ is tangent to $BC$ at P. $C_{2}$ and $C_{3}$ are tangent to big circle and line $BC$ and circle $C_{4}$. [img]http://aycu01.webshots.com/image/4120/2005120338156776027_rs.jpg[/img] Prove : \[r_{1}+r_{2}+r_{3}=R\] ($R$ radius of big circle)

Let there be a given parabola $y^2=4ax$ in the coordinate plane. Consider all chords of the parabola that are visible at a right angle from the origin of the coordinate system. Prove that all these chords pass through a fixed point.

Let $\mathcal P$ be the parabola $y^2=2px$, and let $T_0$ be a point on it. Point $T_0'$ is such that the midpoint of the segment $T_0T_0'$ lies on the axis of the parabola. For a variable point $T$ on $\mathcal P$, the perpendicular from $T_0'$ to the line $T_0T$ intersects the line through $T$ parallel to the axis of $\mathcal P$ at a point $T'$. Find the locus of $T'$.

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.

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 square $ABCD$ is divided into $(n - 1)^2$ congruent squares, with sides parallel to the sides of the given square. Consider the grid of all $n^2$ corners obtained in this manner. Determine all integers $n$ for which it is possible to construct a non-degenerate parabola with its axis parallel to one side of the square and that passes through exactly $n$ points of the grid.

A parabola $y = ax^{2} + bx + c$ has vertex $(4,2)$. If $(2,0)$ is on the parabola, then $abc$ equals $ \textbf{(A)}\ -12\qquad\textbf{(B)}\ -6\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 12$

Let $ t$ be a positive number. Draw two tangent lines from the point $ (t, \minus{} 1)$ to the parabpla $ y \equal{} x^2$. Denote $ S(t)$ the area bounded by the tangents line and the parabola. Find the minimum value of $ \frac {S(t)}{\sqrt {t}}$.

$PQ$ is a chord of parabola $y^2=2px(p>0)$ and $PQ$ pass its focus $F$. Line $l$ is its directrix. Projection of $PQ$ on $l$ is $MN$. The area of curved surface that $PQ$ rotate around $l$ is $S_1$, the area of spherical surface of the ball with diameter of $MN$ is $S_2$, then $\text{(A)}S_1>S_2\qquad\text{(B)}S_1<S_2\qquad\text{(C)}S_1\geq S_2\qquad\text{(D)}$ Not sure

Let $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, and $P_6$ be six parabolas in the plane, each congruent to the parabola $y = x^2/16$. The vertices of the six parabolas are evenly spaced around a circle. The parabolas open outward with their axes being extensions of six of the circle's radii. Parabola $P_1$ is tangent to $P_2$, which is tangent to $P_3$, which is tangent to $P_4$, which is tangent to $P_5$, which is tangent to $P_6$, which is tangent to $P_1$. What is the diameter of the circle?

If ellipse $x^2+4(y-a)^2=4$ and parabola $x^2=2y$ have intersections, then the range value of $a$ is________.

$f_1(x)=x^2+p_1x+q_1,f_2(x)=x^2+p_2x+q_2$ are two parabolas. $l_1$ and $l_2$ are two not parallel lines. It is knows, that segments, that cuted on the $l_1$ by parabolas are equals, and segments, that cuted on the $l_2$ by parabolas are equals too. Prove, that parabolas are equals.

In $ xy$ plane, find the minimum volume of the solid by rotating the region boubded by the parabola $ y \equal{} x^2 \plus{} ax \plus{} b$ passing through the point $ (1,\ \minus{} 1)$ and the $ x$ axis about the $ x$ axis

Is it possible to cover the plane with the interiors of a finite number of parabolas?

In equilateral triangle $ABC$ with side length $2$, let the parabola with focus $A$ and directrix $BC$ intersect sides $AB$ and $AC$ at $A_1$ and $A_2$, respectively. Similarly, let the parabola with focus $B$ and directrix $CA$ intersect sides $BC$ and $BA$ at $B_1$ and $B_2$, respectively. Finally, let the parabola with focus $C$ and directrix $AB$ intersect sides $CA$ and $C_B$ at $C_1$ and $C_2$, respectively. Find the perimeter of the triangle formed by lines $A_1A_2$, $B_1B_2$, $C_1C_2$.

The lines $t$ and $ t'$, tangent to the parabola $y = x^2$ at points $A$ and $B$ respectively, intersect at point $C$. The median of triangle $ABC$ from $C$ has length $m$. Find the area of $\triangle ABC$ in terms of $m$.

For $f(x)=\frac{1}{x}\ (x>0)$, prove the following inequality. \[f\left(t+\frac 12 \right)\leq \int_t^{t+1} f(x)\ dx\leq \frac 16\left\{f(t)+4f\left(t+\frac 12\right)+f(t+1)\right\}\]

A spinning turntable is rotating in a vertical plane with period $ 500 \, \text{ms} $. It has diameter 2 feet carries a ping-pong ball at the edge of its circumference. The ball is bolted on to the turntable but is released from its clutch at a moment in time when the ball makes a subtended angle of $\theta>0$ with the respect to the horizontal axis that crosses the center. This is illustrated in the figure. The ball flies up in the air, making a parabola and, when it comes back down, it does not hit the turntable. This can happen only if $\theta>\theta_m$. Find $\theta_m$, rounded to the nearest integer degree? [asy] filldraw(circle((0,0),1),gray(0.7)); draw((0,0)--(0.81915, 0.57358)); dot((0.81915, 0.57358)); draw((0.81915, 0.57358)--(0.475006, 1.06507)); arrow((0.417649,1.14698), dir(305), 12); draw((0,0)--(1,0),dashed); label("$\theta$", (0.2, 0.2/3), fontsize(8)); label("$r$", (0.409575,0.28679), NW, fontsize(8)); [/asy] [i]Problem proposed by Ahaan Rungta[/i]

A circle centered at a point $F$ and a parabola with focus $F$ have two common points. Prove that there exist four points $A, B, C, D$ on the circle such that the lines $AB, BC, CD$ and $DA$ touch the parabola.

Let $ABCD$ be a square such that $AB$ lies along the line $y=x+8,$ and $C$ and $D$ lie on the parabola $y=x^2.$ Find all possible values of sidelength of the square.