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.

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}$.

Find the equation of the line tangent to the parabola $y = 1/3(x^2-2x+4)$ and a circle of unit radius centered at the origin. (List all solutions.)

The parabola $ y \equal{} ax^2 \plus{} bx \plus{} c$ has vertex $ (p,p)$ and $ y$-intercept $ (0, \minus{} p)$, where $ p\neq 0$. What is $ b$? $ \textbf{(A) } \minus{} p \qquad \textbf{(B) } 0 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } p$

Find the positive value of $ a$ such that the curve $ C_1: x \equal{} \sqrt {2y^2 \plus{} \frac {25}{2}}$ tangent to the parabola $ C_2: y \equal{} ax^2$, then find the equation of the tangent line of $ C_1$ at the point of tangency.

The vertices of the convex quadrilateral $ABCD$ lie on the parabola $y=x^2$. It is known that $ABCD$ is cyclic and $AC$ is a diameter of its circumcircle. Let $M$ and $N$ be the midpoints of the diagonals of $AC$ and $BD$ respectively. Find the length of the projection of the segment $MN$ on the axis $Oy$.

Let points $A, B$ and $C$ lie on the parabola $\Delta$ such that the point $H$, orthocenter of triangle $ABC$, coincides with the focus of parabola $\Delta$. Prove that by changing the position of points $A, B$ and $C$ on $\Delta$ so that the orthocenter remain at $H$, inradius of triangle $ABC$ remains unchanged. [i]Proposed by Mahdi Etesamifard[/i]

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]

Define a parabola $C$ by $y=x^2+1$ on the coordinate plane. Let $s,\ t$ be real numbers with $t<0$. Denote by $l_1,\ l_2$ the tangent lines drawn from the point $(s,\ t)$ to the parabola $C$. (1) Find the equations of the tangents $l_1,\ l_2$. (2) Let $a$ be positive real number. Find the pairs of $(s,\ t)$ such that the area of the region enclosed by $C,\ l_1,\ l_2$ is $a$.

$A(1,1)$ is a point on parabola $y=x^2$. Draw the tangent line of the parabola that passes $A$, the line intersects $x$-axis at $D$, intersects $y$-axis at $B$. $C$ is a point on the parabola, and $E$ is a point on segment $AC$, such that $\frac{AE}{EC}=\lambda_1$, $F$ is a point on segment $BC$, such that $\frac{BF}{FC}=\lambda_2$. If $\lambda_1+\lambda_2=1$, $CD$ and $EF$ intersect at $P$. When $C$ moves, find the path equation of $P$.

Let $a,\ b$ be real numbers. Suppose that a function $f(x)$ satisfies $f(x)=a\sin x+b\cos x+\int_{-\pi}^{\pi} f(t)\cos t\ dt$ and has the maximum value $2\pi$ for $-\pi \leq x\leq \pi$. Find the minimum value of $\int_{-\pi}^{\pi} \{f(x)\}^2dx.$ [i]2010 Chiba University entrance exam[/i]

Let $A, B$ be two variable points on a parabola $P_{0}$, such that the tangents at $A$ and $B$ are perpendicular to each other. Show that the locus of the centroid of the triangle formed by $A,B$ and the vertex of $P_0$ is a parabola $P_1 .$ Apply the same process to $P_1$ and repeat the process, obtaining the sequence of parabolas $P_1, P_2 , \ldots, P_n$. If the equation of $P_0$ is $y=m x^2$, find the equation of $P_n .$

The parabola with equation $y = x^2 - 4$ is rotated $60^\circ$ counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has $y$-coordinate $\frac{a - \sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers, and $a$ and $c$ are relatively prime. Find $a + b + c$.

Let $ \mathcal P$ be a parabola, and let $ V_1$ and $ F_1$ be its vertex and focus, respectively. Let $ A$ and $ B$ be points on $ \mathcal P$ so that $ \angle AV_1 B \equal{} 90^\circ$. Let $ \mathcal Q$ be the locus of the midpoint of $ AB$. It turns out that $ \mathcal Q$ is also a parabola, and let $ V_2$ and $ F_2$ denote its vertex and focus, respectively. Determine the ratio $ F_1F_2/V_1V_2$.

Find all pairs $(x, y)$ of real numbers such that \[y^2 - [x]^2 = 19.99 \text{ and } x^2 + [y]^2 = 1999\] where $f(x)=[x]$ is the floor function.

Given a parabola $C : y=1-x^2$ in $xy$-palne with the origin $O$. Take two points $P(p,\ 1-p^2),\ Q(q,\ 1-q^2)\ (p<q)$ on $C$. (1) Express the area $S$ of the part enclosed by two segments $OP,\ OQ$ and the parabalola $C$ in terms of $p,\ q$. (2) If $q=p+1$, then find the minimum value of $S$. (3) If $pq=-1$, then find the minimum value of $S$.

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.

The expression $ x^3 \plus{} . . . x^2 \plus{} . . . x \plus{} ... \equal{} 0$ is written on the blackboard. Two pupils alternately replace the dots by real numbers. The first pupil attempts to obtain an equation having exactly one real root. Can his opponent spoil his efforts?

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.

Find all positive values of $a$, for which there is a number $b$ such that the parabola $y=ax^2-b$ intersects the unit circle at four distinct points. Also prove that for every such a there exists $b$ such that the parabola $y=ax^2-b$ intersects the unit circle at four distinct points whose $x$-coordinates form an arithmetic progression.

A parabola is inscribed in equilateral triangle $ABC$ of side length $1$ in the sense that $AC$ and $BC$ are tangent to the parabola at $A$ and $B$, respectively. Find the area between $AB$ and the parabola.

All three vertices of $\bigtriangleup ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$? $\textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 16$

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.

On the coordinate plane, the parabola $y = x^2$ and the points $A(x_1, x_1^2)$, $B(x_2, x_2^2)$ are set such that $x_1=-998$, $x_2 =1999$ The segments $BX_1$, $AX_2$, $BX_3$, $AX_4$,..., $BX_{1997}$, $AX_{1998}$ and $X_k$ are constructed succesively with $(x_k,0)$, $1 \le k \le 1998$ and $x_3$, $x_4$,..., $x_{1998}$ are abscissas of the points of intersection of the parabola with segments $BX_1$, $AX_2$, $BX_3$, $AX_4$,..., $BX_{1997}$, $AX_{1998}$. Find the value $\frac{1}{x_{1999}}+\frac{1}{x_{2000}}$

Find the volume formed by the revolution of the region satisfying $ 0\leq y\leq (x \minus{} p)(q \minus{} x)\ (0 < p < q)$ in the coordinate plane about the $ y$ -axis. You are not allowed to use the formula: $ V \equal{} \boxed{\int_a^b 2\pi x|f(x)|\ dx\ (a < b)}$ here.