This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 137

1973 Polish MO Finals, 6

Prove that for every centrally symmetric polygon there is at most one ellipse containing the polygon and having the minimal area.

2014-2015 SDML (High School), 8

Tags: conic , ellipse , geometry
What is the maximum area of a triangle that can be inscribed in an ellipse with semi-axes $a$ and $b$? $\text{(A) }ab\frac{3\sqrt{3}}{4}\qquad\text{(B) }ab\qquad\text{(C) }ab\sqrt{2}\qquad\text{(D) }\left(a+b\right)\frac{3\sqrt{3}}{4}\qquad\text{(E) }\left(a+b\right)\sqrt{2}$

2002 Iran MO (3rd Round), 6

$M$ 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)

2022 Sharygin Geometry Olympiad, 23

Tags: ellipse , conic , geometry
An ellipse with focus $F$ is given. Two perpendicular lines passing through $F$ meet the ellipse at four points. The tangents to the ellipse at these points form a quadrilateral circumscribed around the ellipse. Prove that this quadrilateral is inscribed into a conic with focus $F$

PEN H Problems, 4

Find all pairs $(x, y)$ of positive rational numbers such that $x^{2}+3y^{2}=1$.

1997 National High School Mathematics League, 4

Tags: ellipse , conic
In rectangular coordinate system, if $m(x^2+y^2+2y+1)=(x-2y+3)^2$ refers to an ellipse, then the range value of $m$ is $\text{(A)}(0,1)\qquad\text{(B)}(1,+\infty)\qquad\text{(C)}(0,5)\qquad\text{(D)}(5,+\infty)$

1996 IMC, 10

Tags: conic , geometry , ellipse
Let $B$ be a bounded closed convex symmetric (with respect to the origin) set in $\mathbb{R}^{2}$ with boundary $\Gamma$. Let $B$ have the property that the ellipse of maximal area contained in $B$ is the disc $D$ of radius $1$ centered at the origin with boundary $C$. Prove that $A \cap \Gamma \ne \emptyset$ for any arc $A$ of $C$ of length $l(A)\geq \frac{\pi}{2}$.

1995 Putnam, 2

An ellipse, whose semi-axes have length $a$ and $b$, rolls without slipping on the curve $y=c\sin{\left(\frac{x}{a}\right)}$. How are $a,b,c$ related, given that the ellipse completes one revolution when it traverses one period of the curve?

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________.

2018 CMIMC Geometry, 9

Tags: conic , ellipse , geometry
Suppose $\mathcal{E}_1 \neq \mathcal{E}_2$ are two intersecting ellipses with a common focus $X$; let the common external tangents of $\mathcal{E}_1$ and $\mathcal{E}_2$ intersect at a point $Y$. Further suppose that $X_1$ and $X_2$ are the other foci of $\mathcal{E}_1$ and $\mathcal{E}_2$, respectively, such that $X_1\in \mathcal{E}_2$ and $X_2\in \mathcal{E}_1$. If $X_1X_2=8, XX_2=7$, and $XX_1=9$, what is $XY^2$?

2006 China Second Round Olympiad, 9

Tags: conic , ellipse , ratio , geometry
Suppose points $F_1, F_2$ are the left and right foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{4}=1$ respectively, and point $P$ is on line $l:$, $x-\sqrt{3} y+8+2\sqrt{3}=0$. Find the value of ratio $\frac{|PF_1|}{|PF_2|}$ when $\angle F_1PF_2$ reaches its maximum value.

2001 Cuba MO, 4

Tags: tangent , ellipse , conic
The tangents at four different points of an arc of a circle less than $180^o$ intersect forming a convex quadrilateral $ABCD$. Prove that two of the vertices belong to an ellipse whose foci to the other two vertices.

1973 IMO Longlists, 4

Tags: conic , ellipse , geometry
A circle of radius 1 is placed in a corner of a room (i.e., it touches the horizontal floor and two vertical walls perpendicular to each other). Find the locus of the center of the band for all of its possible positions. [b]Note.[/b] For the solution of this problem, it is useful to know the following Monge theorem: The locus of all points $P$, such that the two tangents from $P$ to the ellipse with equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are perpendicular to each other, is a circle − a so-called Monge circle − with equation $x^2 + y^2 = a^2 + b^2$.

2013 Today's Calculation Of Integral, 870

Consider the ellipse $E: 3x^2+y^2=3$ and the hyperbola $H: xy=\frac 34.$ (1) Find all points of intersection of $E$ and $H$. (2) Find the area of the region expressed by the system of inequality \[\left\{ \begin{array}{ll} 3x^2+y^2\leq 3 &\quad \\ xy\geq \frac 34 , &\quad \end{array} \right.\]

2020 Purple Comet Problems, 24

Tags: ellipse
Points $E$ and $F$ lie on diagonal $\overline{AC}$ of square $ABCD$ with side length $24$, such that $AE = CF = 3\sqrt2$. An ellipse with foci at $E$ and $F$ is tangent to the sides of the square. Find the sum of the distances from any point on the ellipse to the two foci.

2005 AIME Problems, 15

Let $w_{1}$ and $w_{2}$ denote the circles $x^{2}+y^{2}+10x-24y-87=0$ and $x^{2}+y^{2}-10x-24y+153=0$, respectively. Let $m$ be the smallest positive value of $a$ for which the line $y=ax$ contains the center of a circle that is externally tangent to $w_{2}$ and internally tangent to $w_{1}$. Given that $m^{2}=p/q$, where $p$ and $q$ are relatively prime integers, find $p+q$.

1999 Federal Competition For Advanced Students, Part 2, 2

Let $\epsilon$ be a plane and $k_1, k_2, k_3$ be spheres on the same side of $\epsilon$. The spheres $k_1, k_2, k_3$ touch the plane at points $T_1, T_2, T_3$, respectively, and $k_2$ touches $k_1$ at $S_1$ and $k_3$ at $S_3$. Prove that the lines $S_1T_1$ and $S_3T_3$ intersect on the sphere $k_2$. Describe the locus of the intersection point.

2012 ELMO Shortlist, 5

Let $ABC$ be an acute triangle with $AB<AC$, and let $D$ and $E$ be points on side $BC$ such that $BD=CE$ and $D$ lies between $B$ and $E$. Suppose there exists a point $P$ inside $ABC$ such that $PD\parallel AE$ and $\angle PAB=\angle EAC$. Prove that $\angle PBA=\angle PCA$. [i]Calvin Deng.[/i]

2024 Sharygin Geometry Olympiad, 22

A segment $AB$ is given. Let $C$ be an arbitrary point of the perpendicular bisector to $AB$; $O$ be the point on the circumcircle of $ABC$ opposite to $C$; and an ellipse centred at $O$ touch $AB, BC, CA$. Find the locus of touching points of the ellipse with the line $BC$.

1972 Putnam, A4

Tags: ellipse , square
Show that a circle inscribed in a square has a larger perimeter than any other ellipse inscribed in the square.

1986 AMC 12/AHSME, 18

Tags: conic , ellipse , geometry
A plane intersects a right circular cylinder of radius $1$ forming an ellipse. If the major axis of the ellipse of $50\%$ longer than the minor axis, the length of the major axis is $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{3}{2}\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ \frac{9}{4}\qquad\textbf{(E)}\ 3$

2009 Today's Calculation Of Integral, 493

In the $ x \minus{} y$ plane, let $ l$ be the tangent line at the point $ A\left(\frac {a}{2},\ \frac {\sqrt {3}}{2}b\right)$ on the ellipse $ \frac {x^2}{a^2} \plus{} \frac {y^2}{b^2}\equal{}1\ (0 < b < 1 < a)$. Let denote $ S$ be the area of the figure bounded by $ l,$ the $ x$ axis and the ellipse. (1) Find the equation of $ l$. (2) Express $ S$ in terms of $ a,\ b$. (3) Find the maximum value of $ S$ with the constraint $ a^2 \plus{} 3b^2 \equal{} 4$.

2000 Harvard-MIT Mathematics Tournament, 7

Tags: ellipse , geometry , conic
Let $ABC$ be a triangle inscribed in the ellipse $\frac{x^2}{4} +\frac{y^2}{9}= 1$. If its centroid is the origin $(0,0)$, find its area.

1979 Spain Mathematical Olympiad, 1

Calculate the area of the intersection of the interior of the ellipse $\frac{x^2}{16}+ \frac{y^2}{4}= 1$ with the circle bounded by the circumference $(x -2)^2 + (y - 1)^2 = 5$.

1985 IMO Longlists, 31

Let $E_1, E_2$, and $E_3$ be three mutually intersecting ellipses, all in the same plane. Their foci are respectively $F_2, F_3; F_3, F_1$; and $F_1, F_2$. The three foci are not on a straight line. Prove that the common chords of each pair of ellipses are concurrent.