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

The value of $x^2-6x+13$ can never be less than: $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 4.5 \qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 13 $

Show that the plane cannot be represented as the union of the inner regions of a finite number of parabolas.

Let $ S$ be the set of ordered triples $ (x,y,z)$ of real numbers for which \[ \log_{10} (x \plus{} y) \equal{} z\text{ and }\log_{10} (x^2 \plus{} y^2) \equal{} z \plus{} 1. \]There are real numbers $ a$ and $ b$ such that for all ordered triples $ (x,y,z)$ in $ S$ we have $ x^3 \plus{} y^3 \equal{} a \cdot 10^{3z} \plus{} b \cdot 10^{2z}$. What is the value of $ a \plus{} b$? $ \textbf{(A)}\ \frac {15}{2}\qquad \textbf{(B)}\ \frac {29}{2}\qquad \textbf{(C)}\ 15\qquad \textbf{(D)}\ \frac {39}{2}\qquad \textbf{(E)}\ 24$

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.

The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$. For how many points $(x,y)\in P$ with integer coefficients is it true that $|4x+3y|\leq 1000$? $\textbf{(A) }38\qquad \textbf{(B) }40\qquad \textbf{(C) }42\qquad \textbf{(D) }44\qquad \textbf{(E) }46\qquad$

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

$(BEL 1)$ A parabola $P_1$ with equation $x^2 - 2py = 0$ and parabola $P_2$ with equation $x^2 + 2py = 0, p > 0$, are given. A line $t$ is tangent to $P_2.$ Find the locus of pole $M$ of the line $t$ with respect to $P_1.$

Let $z$ be a complex number and k a positive integer such that $z^k$ is a positive real number other than $1$. Let $f(n)$ denote the real part of the complex number $z^n$. Assume the parabola $p(n) = an^2 +bn+c$ intersects $f(n)$ four times, at $n = 0, 1, 2, 3$. Assuming the smallest possible value of $k$, find the largest possible value of $a$.

On the plane, a Cartesian coordinate system is chosen. Given points $ A_1,A_2,A_3,A_4$ on the parabola $ y \equal{} x^2$, and points $ B_1,B_2,B_3,B_4$ on the parabola $ y \equal{} 2009x^2$. Points $ A_1,A_2,A_3,A_4$ are concyclic, and points $ A_i$ and $ B_i$ have equal abscissas for each $ i \equal{} 1,2,3,4$. Prove that points $ B_1,B_2,B_3,B_4$ are also concyclic.

A segment through the focus $F$ of a parabola with vertex $V$ is perpendicular to $\overline{FV}$ and intersects the parabola in points $A$ and $B$. What is $\cos(\angle AVB)$? $ \textbf{(A)}\ -\frac{3\sqrt{5}}{7} \qquad \textbf{(B)}\ -\frac{2\sqrt{5}}{5} \qquad \textbf{(C)}\ -\frac{4}{5} \qquad \textbf{(D)}\ -\frac{3}{5} \qquad \textbf{(E)}\ -\frac{1}{2} $

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 $\text{(A)}\frac{1991}{1992}\qquad\text{(B)}\frac{1992}{1993}\qquad\text{(C)}\frac{1991}{1993}\qquad\text{(D)}\frac{1993}{1992}$

$AB$ and $CD$ are two parallel chords of a parabola. Circle $S_1$ passing through points $A,B$ intersects circle $S_2$ passing through $C,D$ at points $E,F$. Prove that if $E$ belongs to the parabola, then $F$ also belongs to the parabola. I.Voronovich

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 $P(P=l\cap m)$.

Suppose a parabola with the axis as the $ y$ axis, concave up and touches the graph $ y\equal{}1\minus{}|x|$. Find the equation of the parabola such that the area of the region surrounded by the parabola and the $ x$ axis is maximal.

Answer either (i) or (ii): (i) Let $a>0.$ Three straight lines pass through the three points $(0,-a,a), (a,0,-a)$ and $(-a,a,0),$ parallel to the $x-,y-$ and $z-$axis, respectively. A variable straight line moves so that it has one point in common with each of the three given lines. Find the equation of the surface described by the variable line. (II) Which planes cut the surface $xy+yz+xz=0$ in (1) circles, (2) parabolas?

$A,B,C$ 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.

Let $ a,\ b$ be the real numbers such that $ 0\leq a\leq b\leq 1$. Find the minimum value of $ \int_0^1 |(x\minus{}a)(x\minus{}b)|\ dx$.

$M$ 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)

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 $\textbf{(A) }\text{the line segment from }A\text{ to }B\qquad$ $\textbf{(B) }\text{the line passing through }A\text{ and }B\qquad$ $\textbf{(C) }\text{the perpendicular bisector of the line segment from }A\text{ to }B\qquad$ $\textbf{(D) }\text{an elllipse having positive area}\qquad$ $\textbf{(E) }\text{a parabola}$

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 $m$.

A triangle $ABC$ is inscribed in a circle $\omega$. A variable line $\ell$ chosen parallel to $BC$ meets segments $AB$, $AC$ at points $D$, $E$ respectively, and meets $\omega$ at points $K$, $L$ (where $D$ lies between $K$ and $E$). Circle $\gamma_1$ is tangent to the segments $KD$ and $BD$ and also tangent to $\omega$, while circle $\gamma_2$ is tangent to the segments $LE$ and $CE$ and also tangent to $\omega$. Determine the locus, as $\ell$ varies, of the meeting point of the common inner tangents to $\gamma_1$ and $\gamma_2$. [i](Russia) Vasily Mokin and Fedor Ivlev[/i]

Given a line segment $ PQ$ moving on the parabola $ y \equal{} x^2$ with end points on the parabola. The area of the figure surrounded by $ PQ$ and the parabola is always equal to $ \frac {4}{3}$. Find the equation of the locus of the mid point $ M$ of $ PQ$.

$C$ is the circle center $(0,1)$, radius $1$. $P$ is the parabola $y = ax^2$. They meet at $(0, 0)$. For what values of $a$ do they meet at another point or points?

When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio $2$ to $1$. How many integer values of $k$ are there such that $0<k\leq 2007$ and the area between the parabola $y=k-x^2$ and the $x$-axis is an integer? [asy] import graph; size(300); defaultpen(linewidth(0.8)+fontsize(10)); real k=1.5; real endp=sqrt(k); real f(real x) { return k-x^2; } path parabola=graph(f,-endp,endp)--cycle; filldraw(parabola, lightgray); draw((endp,0)--(endp,k)--(-endp,k)--(-endp,0)); label("Region I", (0,2*k/5)); label("Box II", (51/64*endp,13/16*k)); label("area(I) = $\frac23$\,area(II)",(5/3*endp,k/2)); [/asy]