Found problems: 475
2006 AMC 12/AHSME, 21
Rectangle $ ABCD$ has area 2006. An ellipse with area $ 2006\pi$ passes through $ A$ and $ C$ and has foci at $ B$ and $ D$. What is the perimeter of the rectangle? (The area of an ellipse is $ \pi ab$, where $ 2a$ and $ 2b$ are the lengths of its axes.)
$ \textbf{(A) } \frac {16\sqrt {2006}}{\pi} \qquad \textbf{(B) } \frac {1003}4 \qquad \textbf{(C) } 8\sqrt {1003} \qquad \textbf{(D) } 6\sqrt {2006} \qquad \textbf{(E) } \frac {32\sqrt {1003}}\pi$
1984 National High School Mathematics League, 3
For any integers $1\leq n\leq m\leq5$, how many hyperbolas does the equation $\rho=\frac{1}{1-\text{C}_m^n \cos\theta}$ represent?
Note: $\text{C}_m^n=\frac{m!}{n!(m-n)!}$.
$\text{(A)}15\qquad\text{(B)}10\qquad\text{(C)}7\qquad\text{(D)}6$
1990 Bulgaria National Olympiad, Problem 2
Let be given a real number $\alpha\ne0$. Show that there is a unique point $P$ in the coordinate plane, such that for every line through $P$ which intersects the parabola $y=\alpha x^2$ in two distinct points $A$ and $B$, segments $OA$ and $OB$ are perpendicular (where $O$ is the origin).
2006 Romania National Olympiad, 2
Let $n$ be a positive integer. Prove that there exists an integer $k$, $k\geq 2$, and numbers $a_i \in \{ -1, 1 \}$, such that \[ n = \sum_{1\leq i < j \leq k } a_ia_j . \]
2015 ISI Entrance Examination, 2
Let $y = x^2 + ax + b$ be a parabola that cuts the coordinate axes at three distinct points. Show that the circle passing through these three points also passes through $(0,1)$.
Kvant 2024, M2823
A parabola $p$ is drawn on the coordinate plane — the graph of the equation $y =-x^2$, and a point $A$ is marked that does not lie on the parabola $p$. All possible parabolas $q$ of the form $y = x^2+ax+b$ are drawn through point $A$, intersecting $p$ at two points $X$ and $Y$ . Prove that all possible $XY$ lines pass through a fixed point in the plane.
[i]P.A.Kozhevnikov[/i]
1969 IMO Longlists, 1
$(BEL 1)$ 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 $P_1.$
1986 National High School Mathematics League, 7
Inside a circular column with a bottom surface with radius of $6$, there are two balls with radius of $6$ as well. The distance betwen their centers is $13$. Draw a plane that is tangent to both spherical surface, intersect the circular column at a curve $C$. $C$ is a ellipse, then the sum of its short axis and long axis is________.
1998 National High School Mathematics League, 11
If ellipse $x^2+4(y-a)^2=4$ and parabola $x^2=2y$ have intersections, then the range value of $a$ is________.
2009 Today's Calculation Of Integral, 506
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$.
2021 Math Prize for Girls Problems, 12
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?
2011 Today's Calculation Of Integral, 769
In $xyz$ space, find the volume of the solid expressed by $x^2+y^2\leq z\le \sqrt{3}y+1.$
2011 Today's Calculation Of Integral, 689
Let $C: y=x^2+ax+b$ be a parabola passing through the point $(1,\ -1)$. Find the minimum volume of the figure enclosed by $C$ and the $x$ axis by a rotation about the $x$ axis.
Proposed by kunny
2006 Belarusian National Olympiad, 5
A convex quadrilateral $ABCD$ Is placed on the Cartesian plane. Its vertices $A$ and $D$ belong to the negative branch of the graph of the hyperbola $y= 1/x$, the vertices $B$ and $C$ belong to the positive branch of the graph and point $B$ lies at the left of $C$, the segment $AC$ passes through the origin $(0,0)$. Prove that $\angle BAD = \angle BCD$.
(I, Voronovich)
2017 BMT Spring, 19
Let $T$ be the triangle in the $xy$-plane with vertices $(0, 0)$, $(3, 0)$, and $\left(0, \frac32\right)$. Let $E$ be the ellipse inscribed in $T$ which meets each side of $T$ at its midpoint. Find the distance from the center of $E$ to $(0, 0)$.
2023 Assara - South Russian Girl's MO, 7
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.
2012 ISI Entrance Examination, 6
[b]i)[/b] Let $0<a<b$.Prove that amongst all triangles having base $a$ and perimeter $a+b$ the triangle having two sides(other than the base) equal to $\frac {b}{2}$ has the maximum area.
[b]ii)[/b]Using $i)$ or otherwise, prove that amongst all quadrilateral having give perimeter the square has the maximum area.
2005 Romania Team Selection Test, 2
Let $ABC$ be a triangle, and let $D$, $E$, $F$ be 3 points on the sides $BC$, $CA$ and $AB$ respectively, such that the inradii of the triangles $AEF$, $BDF$ and $CDE$ are equal with half of the inradius of the triangle $ABC$. Prove that $D$, $E$, $F$ are the midpoints of the sides of the triangle $ABC$.
2012 AIME Problems, 6
The complex numbers $z$ and $w$ satisfy $z^{13} = w$, $w^{11} = z$, and the imaginary part of $z$ is $\sin\left(\frac{m\pi}n\right)$ for relatively prime positive integers $m$ and $n$ with $m < n$. Find $n$.
2024 ELMO Shortlist, G8
Let $ABC$ be a triangle, and let $D$ be a point on the internal angle bisector of $BAC$. Let $x$ be the ellipse with foci $B$ and $C$ passing through $D$, $y$ be the ellipse with foci $A$ and $C$ passing through $D$, and $z$ be the ellipse with foci $A$ and $B$ passing through $D$. Ellipses $x$ and $z$ intersect at distinct points $D$ and $E$, and ellipses $x$ and $y$ intersect at distinct points $D$ and $F$. Prove that $AD$ bisects angle $EAF$.
[i]Andrew Carratu[/i]
2013 Hitotsubashi University Entrance Examination, 3
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 $S$.
2024 Oral Moscow Geometry Olympiad, 1
In a plane:
1. An ellipse with foci $F_1$, $F_2$ lies inside a circle $\omega$. Construct a chord $AB$ of $\omega$. touching the ellipse and such that $A$, $B$, $F_1$, and $F_2$ are concyclic.
2. Let a point $P$ lie inside an acute angled triangle $ABC$, and $A'$, $B'$, $C'$ be the projections of $P$ to $BC$, $CA$, $AB$ respectively. Prove that the diameter of circle $A'B'C'$ equals $CP$ if and only if the circle $ABP$ passes through the circumcenter of $ABC$.
[i]Proposed by Alexey Zaslavsky[/i]
[img]https://cdn.artofproblemsolving.com/attachments/8/e/ac4a006967fb7013efbabf03e55a194cbaa18b.png[/img]
2007 Purple Comet Problems, 20
Three congruent ellipses are mutually tangent. Their major axes are parallel. Two of the ellipses are tangent at the end points of their minor axes as shown. The distance between the centers of these two ellipses is $4$. The distances from those two centers to the center of the third ellipse are both $14$. There are positive integers m and n so that the area between these three ellipses is $\sqrt{n}-m \pi$. Find $m+n$.
[asy]
size(250);
filldraw(ellipse((2.2,0),2,1),grey);
filldraw(ellipse((0,-2),4,2),white);
filldraw(ellipse((0,+2),4,2),white);
filldraw(ellipse((6.94,0),4,2),white);[/asy]
1958 AMC 12/AHSME, 22
A particle is placed on the parabola $ y \equal{} x^2 \minus{} x \minus{} 6$ at a point $ P$ whose $ y$-coordinate is $ 6$. It is allowed to roll along the parabola until it reaches the nearest point $ Q$ whose $ y$-coordinate is $ \minus{}6$. The horizontal distance traveled by the particle (the numerical value of the difference in the $ x$-coordinates of $ P$ and $ Q$) is:
$ \textbf{(A)}\ 5\qquad
\textbf{(B)}\ 4\qquad
\textbf{(C)}\ 3\qquad
\textbf{(D)}\ 2\qquad
\textbf{(E)}\ 1$
2020-IMOC, G4
Let $I$ be the incenter of triangle $ABC$. Let $BI$ and $AC$ intersect at $E$, and $CI$ and $AB$ intersect at $F$. Suppose that $R$ is another intersection of $\odot (ABC)$ and $\odot (AEF)$. Let $M$ be the midpoint of $BC$, and $P, Q$ are the intersections of $AI, MI$ and $EF$, respectively. Show that $A, P, Q, R$ are concyclic.
(ltf0501).