Olympiad problems

Today's calculation of integrals, 873

Tags: ellipse , conic , calculus , integration , analytic geometry , graphing lines , geometry

Let $a,\ b$ be positive real numbers. Consider the circle $C_1: (x-a)^2+y^2=a^2$ and the ellipse $C_2: x^2+\frac{y^2}{b^2}=1.$ (1) Find the condition for which $C_1$ is inscribed in $C_2$. (2) Suppose $b=\frac{1}{\sqrt{3}}$ and $C_1$ is inscribed in $C_2$. Find the coordinate $(p,\ q)$ of the point of tangency in the first quadrant for $C_1$ and $C_2$. (3) Under the condition in (1), find the area of the part enclosed by $C_1,\ C_2$ for $x\geq p$. 60 point

2011 China Second Round Olympiad, 11

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

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]

1976 Putnam, 4

Tags: conic , ellipse

For a point $P$ on an ellipse, let $d$ be the distance from the center of the ellipse to the line tangent to the ellipse at $P.$ Prove that $(PF_1)(PF_2)d^2$ is constant as $P$ varies on the ellipse, where $PF_1$ and $PF_2$ are distances from $P$ to the foci $F_1$ and $F_2$ of the ellipse.

2009 Spain Mathematical Olympiad, 6

Tags: symmetry , conic , geometry , ellipse , reflection , trapezoid , geometric transformation

Inside a circle of center $ O$ and radius $ r$, take two points $ A$ and $ B$ symmetrical about $ O$. We consider a variable point $ P$ on the circle and draw the chord $ \overline{PP'}\perp \overline{AP}$. Let $ C$ is the symmetric of $ B$ about $ \overline{PP'}$ ($ \overline{PP}'$ is the axis of symmetry) . Find the locus of point $ Q \equal{} \overline{PP'}\cap\overline{AC}$ when we change $ P$ in the circle.

2011 ELMO Shortlist, 4

Tags: ellipse , conic , modular arithmetic , geometry , reflection , geometric transformation

Prove that for any convex pentagon $A_1A_2A_3A_4A_5$, there exists a unique pair of points $\{P,Q\}$ (possibly with $P=Q$) such that $\measuredangle{PA_i A_{i-1}} = \measuredangle{A_{i+1}A_iQ}$ for $1\le i\le 5$, where indices are taken $\pmod5$ and angles are directed $\pmod\pi$. [i]Calvin Deng.[/i]

2012 AMC 12/AHSME, 10

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

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} $

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.

1940 Putnam, A5

Tags: conic , ellipse

Prove that the simultaneous equations $$x^4 -x^2 =y^4 -y^2 =z^4 -z^2$$ are satisfied by the points of $4$ straight lines and $6$ ellipses, and by no other points.

2001 AIME Problems, 5

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

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 $m+n.$

1970 AMC 12/AHSME, 10

Tags: ellipse , geometric series , conic , geometry

Let $F=.48181\cdots$ be an infinite repeating decimal with the digits $8$ and $1$ repeating. When $F$ is written as a fraction in lowest terms, the denominator exceeds the numerator by $\textbf{(A) }13\qquad\textbf{(B) }14\qquad\textbf{(C) }29\qquad\textbf{(D) }57\qquad \textbf{(E) }126$

2011 Olympic Revenge, 3

Tags: conic , ellipse , analytic geometry , combinatorics

Let $E$ to be an infinite set of congruent ellipses in the plane, and $r$ a fixed line. It is known that each line parallel to $r$ intersects at least one ellipse belonging to $E$. Prove that there exist infinitely many triples of ellipses belonging to $E$, such that there exists a line that intersect the triple of ellipses.

Today's calculation of integrals, 769

Tags: algebra , integration , calculus , domain , ellipse , conic , function

In $xyz$ space, find the volume of the solid expressed by $x^2+y^2\leq z\le \sqrt{3}y+1.$

2013 Miklós Schweitzer, 11

Tags: advanced fields , ellipse , conic

[list] (a) Consider an ellipse in the plane. Prove that there exists a Riemannian metric which is defined on the whole plane, and with respect to which the ellipse is a geodesic. Prove that the Gaussian curvature of any such Riemannian metric takes a positive value. (b) Consider two nonintersecting, simple closed smooth curves in the plane. Prove that if there is a Riemmanian metric defined on the whole plane and the two curves are geodesics of that metric, then the Gaussian curvature of the metric vanishes somewhere. [/list] [i]Proposed by Tran Quoc Binh[/i]

2008 Harvard-MIT Mathematics Tournament, 12

Tags: ratio , ellipse , 3d geometry , sphere , conic , geometric series , geometry

Suppose we have an (infinite) cone $ \mathcal C$ with apex $ A$ and a plane $ \pi$. The intersection of $ \pi$ and $ \mathcal C$ is an ellipse $ \mathcal E$ with major axis $ BC$, such that $ B$ is closer to $ A$ than $ C$, and $ BC \equal{} 4$, $ AC \equal{} 5$, $ AB \equal{} 3$. Suppose we inscribe a sphere in each part of $ \mathcal C$ cut up by $ \mathcal E$ with both spheres tangent to $ \mathcal E$. What is the ratio of the radii of the spheres (smaller to larger)?

2012 Today's Calculation Of Integral, 854

Tags: analytic geometry , ellipse , conic , geometry , integration , limit , calculus

Given a figure $F: x^2+\frac{y^2}{3}=1$ on the coordinate plane. Denote by $S_n$ the area of the common part of the $n+1' s$ figures formed by rotating $F$ of $\frac{k}{2n}\pi\ (k=0,\ 1,\ 2,\ \cdots,\ n)$ radians counterclockwise about the origin. Find $\lim_{n\to\infty} S_n$.

2006 Taiwan National Olympiad, 3

Tags: analytic geometry , ellipse , slope , conic , graphing lines , geometric transformation , geometry

Let the major axis of an ellipse be $AB$, let $O$ be its center, and let $F$ be one of its foci. $P$ is a point on the ellipse, and $CD$ a chord through $O$, such that $CD$ is parallel to the tangent of the ellipse at $P$. $PF$ and $CD$ intersect at $Q$. Compare the lengths of $PQ$ and $OA$.

2007 Hungary-Israel Binational, 2

Tags: ellipse , rotation , geometry , 3d geometry , geometric transformation , conic , sphere

Given is an ellipse $ e$ in the plane. Find the locus of all points $ P$ in space such that the cone of apex $ P$ and directrix $ e$ is a right circular cone.

1988 National High School Mathematics League, 2

Tags: ellipse , conic

If the coordinate origin is inside the ellipse $k^2x^2+y^2-4kx+2ky+k^2-1=0$, then the range value of $k$ is $\text{(A)}|k|>1\qquad\text{(B)}|k|\neq1\qquad\text{(C)}-1<k<1\qquad\text{(D)}0<|k|<1$

2020 JHMT, 3

Tags: ellipse , conic , geometry

Consider a right cylinder with height $5\sqrt3$. A plane intersects each of the bases of the cylinder at exactly one point, and the cylindric section (the intersection of the plane and the cylinder) forms an ellipse. Find the product of the sum and the dierence of the lengths of the major and minor axes of this ellipse. [i]Note:[/i] An ellipse is a regular oval shape resulting when a cone is cut by an oblique plane which does not intersect the base. The major axis is the longer diameter and the minor axis the shorter.

1985 AIME Problems, 11

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

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?

2004 AMC 12/AHSME, 21

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

The graph of $ 2x^2 \plus{} xy \plus{} 3y^2 \minus{} 11x \minus{} 20y \plus{} 40 \equal{} 0$ is an ellipse in the ﬁrst quadrant of the $ xy$-plane. Let $ a$ and $ b$ be the maximum and minimum values of $ \frac {y}{x}$ over all points $ (x, y)$ on the ellipse. What is the value of $ a \plus{} b$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ \sqrt {10} \qquad \textbf{(C)}\ \frac72 \qquad \textbf{(D)}\ \frac92 \qquad \textbf{(E)}\ 2\sqrt {14}$

2012 ISI Entrance Examination, 7

Tags: ellipse , conic , geometry

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 $\Gamma$

Today's calculation of integrals, 854

Tags: ellipse , conic , integration , limit , geometry , calculus , analytic geometry

Given a figure $F: x^2+\frac{y^2}{3}=1$ on the coordinate plane. Denote by $S_n$ the area of the common part of the $n+1' s$ figures formed by rotating $F$ of $\frac{k}{2n}\pi\ (k=0,\ 1,\ 2,\ \cdots,\ n)$ radians counterclockwise about the origin. Find $\lim_{n\to\infty} S_n$.

2010 AMC 12/AHSME, 19

Tags: ellipse , conic , inequalities , probability

Each of 2010 boxes in a line contains a single red marble, and for $ 1 \le k \le 2010$, the box in the $ kth$ position also contains $ k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $ P(n)$ be the probability that Isabella stops after drawing exactly $ n$ marbles. What is the smallest value of $ n$ for which $ P(n) < \frac {1}{2010}$? $ \textbf{(A)}\ 45 \qquad \textbf{(B)}\ 63 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 201 \qquad \textbf{(E)}\ 1005$

1996 National High School Mathematics League, 1

Tags: conic , ellipse

Connect the commom points of circle$x^2+(y-1)^2=1$ and ellipse $9x^2+(y+1)^2=9$ with line segments, the figure is a $\text{(A)}$ line segment $\text{(B)}$ scalene triangle $\text{(C)}$ equilateral triangle $\text{(D)}$ quadrilateral