Found problems: 253
The line $x-2y-1=0$ insects the parabola $y^2=4x$ at two different points $A, B$. Let $C$ be a point on the parabola such that $\angle ACB=\frac{\pi}{2}$. Find the coordinate of point $C$.
The area of a circle centered at the origin, which is inscribed in the parabola $y=x^2-25$, can be expressed as $\tfrac ab\pi$, where $a$ and $b$ are coprime positive integers. What is the value of $a+b$?
Let $ A$, $ B$, and $ C$ be three distinct points on the graph of $ y\equal{}x^2$ such that line $ AB$ is parallel to the $ x$-axis and $ \triangle{ABC}$ is a right triangle with area $ 2008$. What is the sum of the digits of the $ y$-coordinate of $ C$?
$ \textbf{(A)}\ 16 \qquad
\textbf{(B)}\ 17 \qquad
\textbf{(C)}\ 18 \qquad
\textbf{(D)}\ 19 \qquad
\textbf{(E)}\ 20$
Let $f(x)$ be a function such that every straight line has the same number of intersection points with the graph $y = f(x)$ and with the graph $y = x^2$. Prove that $f(x) = x^2.$
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]
The point $(a,b)$ lies on the circle $x^2+y^2=1$. The tangent to the circle at this point meets the parabola $y=x^2+1$ at exactly one point. Find all such points $(a,b)$.
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.
$A(1,1)$ 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 $l,\ m$ be the tangent lines passing through the point $A(a,\ a-1)$ on the line $y=x-1$ and touch the parabola $y=x^2$.
Note that the slope of $l$ is greater than that of $m$.
(1) Exress the slope of $l$ in terms of $a$.
(2) Denote $P,\ Q$ be the points of tangency of the lines $l,\ m$ and the parabola $y=x^2$.
Find the minimum area of the part bounded by the line segment $PQ$ and the parabola $y=x^2$.
(3) Find the minimum distance between the parabola $y=x^2$ and the line $y=x-1$.
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.
On the graphic of the function $y=x^2$ were selected $1000$ pairwise distinct points, abscissas of which are integer numbers from the segment $[0; 100000]$. Prove that it is possible to choose six different selected points $A$, $B$, $C$, $A'$, $B'$, $C'$ such that areas of triangles $ABC$ and $A'B'C'$ are equals.
[i]A. Tereshin[/i]
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]
If ellipse $x^2+4(y-a)^2=4$ and parabola $x^2=2y$ have intersections, then the range value of $a$ is________.
The graph of $ y \equal{} 2x^2 \plus{} 4x \plus{} 3$ has its:
$ \textbf{(A)}\ \text{lowest point at } {(\minus{}1,9)}\qquad
\textbf{(B)}\ \text{lowest point at } {(1,1)}\qquad \\
\textbf{(C)}\ \text{lowest point at } {(\minus{}1,1)}\qquad
\textbf{(D)}\ \text{highest point at } {(\minus{}1,9)}\qquad \\
\textbf{(E)}\ \text{highest point at } {(\minus{}1,1)}$
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.
Let $ABC$ be a triangle inscribed in the parabola $y=x^2$ such that the line $AB \parallel$ the axis $Ox$. Also point $C$ is closer to the axis $Ox$ than the line $AB$. Given that the length of the segment $AB$ is 1 shorter than the length of the altitude $CH$ (of the triangle $ABC$). Determine the angle $\angle{ACB}$ .
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.
Suppose that a parabola has vertex $\left(\tfrac{1}{4},-\tfrac{9}{8}\right)$, and equation $y=ax^2+bx+c$, where $a>0$ and $a+b+c$ is an integer. The minimum possible value of $a$ can be written as $\tfrac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
Two parabolas have equations $y=x^2+ax+b$ and $y=x^2+cx+d$, where $a$, $b$, $c$, and $d$ are integers (not necessarily different), each chosen independently by rolling a fair six-sided die. What is the probability that the parabolas have at least one point in common?
$\textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{25}{36} \qquad\textbf{(C)}\ \frac{5}{6} \qquad\textbf{(D)}\ \frac{31}{36} \qquad\textbf{(E)}\ 1 $
Points $A(1,2)$, a line that passes $(5,-2)$ intersects the parabola $y^2=4x$ at two points $B,C$. Then, $\triangle ABC$ is
$\text{(A)}$ an acute triangle
$\text{(B)}$ an obtuse triangle
$\text{(C)}$ a right triangle
$\text{(D)}$ not sure
We will say that a point of the plane $(u, v)$ lies between the parabolas $y = f(x)$ and $y = g(x)$ if $f(u) \leq v \leq g(u)$. Find the smallest real $p$ for which the following statement is true: for any segment, the ends and the midpoint of which
lie between the parabolas $y = x^2$ and $y=x^2+1$, then they lie entirely between the parabolas $y=x^2$ and $y=x^2+p$.
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
$\textbf{(A) }\text{the line segment from }A\text{ to }B\qquad$
$\textbf{(B) }\text{the line passing through }A\text{ and }B\qquad$
$\textbf{(C) }\text{the perpendicular bisector of the line segment from }A\text{ to }B\qquad$
$\textbf{(D) }\text{an elllipse having positive area}\qquad$
$\textbf{(E) }\text{a parabola}$
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 $S_1+S_2$.
Consider a parabola. The [i]parabolic length[/i] of a segment is the length of the projection of this segment on a straight line perpendicular to the axis of symmetry of the parabola. In the parabola, two chords $AB$ and $CD$ are drawn, intersecting at the point $N{}$. Prove that the product of the parabolic lengths of the segments $AN$ and $BN$ is equal to the product of the parabolic lengths of the segments $CN$ and $DN$.
[i]Proposed by M. Panov[/i]
Consider the graphs of (1): $ y\equal{}x^2\minus{}\frac{1}{2}x\plus{}2$ and (2) $ y\equal{}x^2\plus{}\frac{1}{2}x\plus{}2$ on the same set of axis. These parabolas are exactly the same shape. Then:
$ \textbf{(A)}\ \text{the graphs coincide.} \\
\textbf{(B)}\ \text{the graph of (1) is lower than the graph of (2).} \\
\textbf{(C)}\ \text{the graph of (1) is to the left of the graph of (2).} \\
\textbf{(D)}\ \text{the graph of (1) is to the right of the graph of (2).} \\
\textbf{(E)}\ \text{the graph of (1) is higher than the graph of (2).}$