Olympiad problems

2009 India Regional Mathematical Olympiad, 5

Tags: inequalities , ellipse , geometry , circumcircle , induction , function , conic

A convex polygon is such that the distance between any two vertices does not exceed $ 1$. $ (i)$ Prove that the distance between any two points on the boundary of the polygon does not exceed $ 1$. $ (ii)$ If $ X$ and $ Y$ are two distinct points inside the polygon, prove that there exists a point $ Z$ on the boundary of the polygon such that $ XZ \plus{} YZ\le1$.

1973 Polish MO Finals, 6

Tags: ellipse , geometry , geometric inequalities , conic

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

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.

2014 District Olympiad, 1

Tags: ellipse , complex number , conic , algebra

Solve for $z\in \mathbb{C}$ the equation : \[ |z-|z+1||=|z+|z-1|| \]

2012 ISI Entrance Examination, 7

Tags: ellipse , geometry , conic

Let $\Gamma_1,\Gamma_2$ be two circles centred at the points $(a,0),(b,0);0<a<b$ and having radii $a,b$ respectively.Let $\Gamma$ be the circle touching $\Gamma_1$ externally and $\Gamma_2$ internally. Find the locus of the centre of of $\Gamma$

1963 Putnam, A6

Tags: ellipse , geometry , conic

Let $U$ and $V$ be any two distinct points on an ellipse, let $M$ be the midpoint of the chord $UV$, and let $AB$ and $CD$ be any two other chords through $M$. If the line $UV$ meets the line $AC$ in the point $P$ and the line $BD$ in the point $Q$, prove that $M$ is the midpoint of the segment $PQ.$

1985 ITAMO, 11

Tags: ellipse , analytic geometry , geometry , geometric transformation , reflection , conic

An ellipse has foci at $(9,20)$ and $(49,55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its major axis?

2001 National High School Mathematics League, 7

Tags: ellipse , conic

The length of minor axis of ellipse $\rho-\frac{1}{2-\cos\theta}$ is________.

2024 ELMO Shortlist, G8

Tags: geometry , ellipse , conic

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]

1991 Arnold's Trivium, 7

Tags: ellipse , conic

How many normals to an ellipse can be drawn from a given point in plane? Find the region in which the number of normals is maximal.

2012 AMC 12/AHSME, 10

Tags: geometry , analytic geometry , quadratic , ellipse , conic

What is the area of the polygon whose vertices are the points of intersection of the curves $x^2+y^2=25$ and $(x-4)^2+9y^2=81$? ${{ \textbf{(A)}\ 24\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 37.5}\qquad\textbf{(E)}\ 42} $

2011 China Second Round Olympiad, 11

Tags: ellipse , analytic geometry , graphing lines , slope , geometry , incenter , conic

A line $\ell$ with slope of $\frac{1}{3}$ insects the ellipse $C:\frac{x^2}{36}+\frac{y^2}{4}=1$ at points $A,B$ and the point $P\left( 3\sqrt{2} , \sqrt{2}\right)$ is above the line $\ell$. [list] [b](1)[/b] Prove that the locus of the incenter of triangle $PAB$ is a segment, [b](2)[/b] If $\angle APB=\frac{\pi}{3}$, then find the area of triangle $PAB$.[/list]

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.

1941 Putnam, A7

Tags: linear algebra , matrix , ellipse , conic

Do either (1) or (2): (1) Prove that the determinant of the matrix $$\begin{pmatrix} 1+a^2 -b^2 -c^2 & 2(ab+c) & 2(ac-b)\\ 2(ab-c) & 1-a^2 +b^2 -c^2 & 2(bc+a)\\ 2(ac+b)& 2(bc-a) & 1-a^2 -b^2 +c^2 \end{pmatrix}$$ is given by $(1+a^2 +b^2 +c^2)^{3}$. (2) A solid is formed by rotating the first quadrant of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ around the $x$-axis. Prove that this solid can rest in stable equilibrium on its vertex if and only if $\frac{a}{b}\leq \sqrt{\frac{8}{5}}$.

2001 AIME Problems, 5

Tags: ellipse , trigonometry , analytic geometry , graphing lines , slope , geometry , conic

An equilateral triangle is inscribed in the ellipse whose equation is $x^2+4y^2=4.$ One vertex of the triangle is $(0,1),$ one altitude is contained in the $y$-axis, and the length of each side is $\sqrt{\frac mn},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

2024 ELMO Shortlist, G8

Tags: geometry , ellipse , conic

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]

1964 AMC 12/AHSME, 2

Tags: parabola , ellipse , conic

The graph of $x^2-4y^2=0$ is: ${{ \textbf{(A)}\ \text{a parabola} \qquad\textbf{(B)}\ \text{an ellipse} \qquad\textbf{(C)}\ \text{a pair of straight lines} \qquad\textbf{(D)}\ \text{a point} }\qquad\textbf{(E)}\ \text{none of these} } $

2023 ELMO Shortlist, G7

Tags: geometry , ellipse

Let $\mathcal E$ be an ellipse with foci $F_1$ and $F_2$, and let $P$ be a point on $\mathcal E$. Suppose lines $PF_1$ and $PF_2$ intersect $\mathcal E$ again at distinct points $A$ and $B$, and the tangents to $\mathcal E$ at $A$ and $B$ intersect at point $Q$. Show that the midpoint of $\overline{PQ}$ lies on the circumcircle of $\triangle PF_1F_2$. [i]Proposed by Karthik Vedula[/i]

2000 China Team Selection Test, 1

Tags: circumcircle , ellipse , angle bisector , geometry , conic

Let $ABC$ be a triangle such that $AB = AC$. Let $D,E$ be points on $AB,AC$ respectively such that $DE = AC$. Let $DE$ meet the circumcircle of triangle $ABC$ at point $T$. Let $P$ be a point on $AT$. Prove that $PD + PE = AT$ if and only if $P$ lies on the circumcircle of triangle $ADE$.

2021 China Second Round Olympiad, Problem 3

Tags: ellipse , complex number , conic

There exists complex numbers $z=x+yi$ such that the point $(x, y)$ lies on the ellipse with equation $\frac{x^2}9+\frac{y^2}{16}=1$. If $\frac{z-1-i}{z-i}$ is real, compute $z$. [i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 3)[/i]

1990 Baltic Way, 9

Tags: ellipse , geometry , congruent triangles , conic

Two congruent triangles are inscribed in an ellipse. Are they necessarily symmetric with respect to an axis or the center of the ellipse?

2009 Today's Calculation Of Integral, 493

Tags: calculus , integration , ellipse , geometry , trigonometry , calculus computations , conic

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$.

1950 AMC 12/AHSME, 49

Tags: ellipse , ratio , trigonometry , conic

A triangle has a fixed base $AB$ that is $2$ inches long. The median from $A$ to side $BC$ is $ 1\frac{1}{2}$ inches long and can have any position emanating from $A$. The locus of the vertex $C$ of the triangle is: $\textbf{(A)}\ \text{A straight line }AB,1\dfrac{1}{2}\text{ inches from }A \qquad\\ \textbf{(B)}\ \text{A circle with }A\text{ as center and radius }2\text{ inches} \qquad\\ \textbf{(C)}\ \text{A circle with }A\text{ as center and radius }3\text{ inches} \qquad\\ \textbf{(D)}\ \text{A circle with radius }3\text{ inches and center }4\text{ inches from }B\text{ along } BA \qquad\\ \textbf{(E)}\ \text{An ellipse with }A\text{ as focus}$

2002 National High School Mathematics League, 4

Tags: ellipse , conic , geometry

Line $\frac{x}{4}+\frac{y}{3}=1$ and ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ intersect at $A$ and $B$. A point on the ellipse $P$ satisties that the area of $\triangle PAB$ is $3$. The number of such points is $\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}3\qquad\text{(D)}4$

2024 Sharygin Geometry Olympiad, 22

Tags: ellipse , geometry , perpendicular bisector , conic

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 $BC$.