Olympiad problems

2000 National High School Mathematics League, 10

Tags: conic , ellipse

In ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $F$ is its left focal point, $A$ is its right vertex, $B$ is its upper vertex. If the eccentricity of the ellipse is $\frac{\sqrt5-1}{2}$, then $\angle ABF=$________.

2003 National High School Mathematics League, 8

Tags: conic , ellipse , geometry

$F_1,F_2$ are two focal points of ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$, $P$ is a point on the ellipse, and $|PF_1|:|PF_2|=2:1$, then the area of $\triangle PF_1F_2$ is________.

2012 Iran MO (3rd Round), 3

Tags: ratio , geometry , geometric transformation , conic , ellipse , dilation , analytic geometry

Cosider ellipse $\epsilon$ with two foci $A$ and $B$ such that the lengths of it's major axis and minor axis are $2a$ and $2b$ respectively. From a point $T$ outside of the ellipse, we draw two tangent lines $TP$ and $TQ$ to the ellipse $\epsilon$. Prove that \[\frac{TP}{TQ}\ge \frac{b}{a}.\] [i]Proposed by Morteza Saghafian[/i]

1941 Putnam, A7

Tags: ellipse , conic , linear algebra , matrix

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 $\frac{a}{b}\leq \sqrt{\frac{8}{5}}$.

2011 Math Prize For Girls Problems, 16

Tags: ellipse , conic , geometry

Let $N$ be the number of ordered pairs of integers $(x, y)$ such that \[ 4x^2 + 9y^2 \le 1000000000. \] Let $a$ be the first digit of $N$ (from the left) and let $b$ be the second digit of $N$. What is the value of $10a + b$ ?

2019 China National Olympiad, 4

Tags: geometry , conic , rhombus , ellipse

Given an ellipse that is not a circle. (1) Prove that the rhombus tangent to the ellipse at all four of its sides with minimum area is unique. (2) Construct this rhombus using a compass and a straight edge.

2011 ITAMO, 4

Tags: conic , ellipse , italy , geometry

$ABCD$ is a convex quadrilateral. $P$ is the intersection of external bisectors of $\angle DAC$ and $\angle DBC$. Prove that $\angle APD = \angle BPC$ if and only if $AD+AC=BC+BD$

2021 China Second Round Olympiad, Problem 12

Tags: conic , ellipse , analytic geometry

Let $C$ be the left vertex of the ellipse $\frac{x^2}8+\frac{y^2}4 = 1$ in the Cartesian Plane. For some real number $k$, the line $y=kx+1$ meets the ellipse at two distinct points $A, B$. (i) Compute the maximum of $|CA|+|CB|$. (ii) Let the line $y=kx+1$ meet the $x$ and $y$ axes at $M$ and $N$, respectively. If the intersection of the perpendicular bisector of $MN$ and the circle with diameter $MN$ lies inside the given ellipse, compute the range of possible values of $k$. [i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 12)[/i]

2008 IMC, 2

Tags: geometry , trigonometry , search , ellipse , conic , analytic geometry , ratio

Two different ellipses are given. One focus of the first ellipse coincides with one focus of the second ellipse. Prove that the ellipses have at most two points in common.

2024 ELMO Shortlist, G8

Tags: conic , ellipse , geometry

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]

2014 Spain Mathematical Olympiad, 3

Tags: conic , parallelogram , ellipse , circumcircle , perpendicular bisector , geometry

Let $B$ and $C$ be two fixed points on a circle centered at $O$ that are not diametrically opposed. Let $A$ be a variable point on the circle distinct from $B$ and $C$ and not belonging to the perpendicular bisector of $BC$. Let $H$ be the orthocenter of $\triangle ABC$, and $M$ and $N$ be the midpoints of the segments $BC$ and $AH$, respectively. The line $AM$ intersects the circle again at $D$, and finally, $NM$ and $OD$ intersect at $P$. Determine the locus of points $P$ as $A$ moves around the circle.

2000 China Team Selection Test, 1

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

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 $ADE$.

1998 IberoAmerican Olympiad For University Students, 2

Tags: conic , ellipse , geometry

In a plane there is an ellipse $E$ with semiaxis $a,b$. Consider all the triangles inscribed in $E$ such that at least one of its sides is parallel to one of the axis of $E$. Find both the locus of the centroid of all such triangles and its area.

Revenge EL(S)MO 2024, 2

Tags: conic , geometry , ellipse

Prove that for any convex quadrilateral there exist an inellipse and circumellipse which are homothetic. Proposed by [i]Benny Wang + Oron Wang[/i]

1963 Putnam, A6

Tags: conic , ellipse , geometry

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 $PQ.$

2021 China Second Round Olympiad, Problem 3

Tags: conic , ellipse , complex number

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]

1970 Canada National Olympiad, 8

Tags: conic , ellipse , parameterization , geometry

Consider all line segments of length 4 with one end-point on the line $y=x$ and the other end-point on the line $y=2x$. Find the equation of the locus of the midpoints of these line segments.

2007 Moldova Team Selection Test, 1

Tags: parabola , symmetry , conic , ellipse , hyperbola , geometry

Show that the plane cannot be represented as the union of the inner regions of a finite number of parabolas.

2012 ISI Entrance Examination, 6

Tags: inequalities , conic , ellipse , calculus , perimeter , perpendicular bisector , geometry

[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.

1971 AMC 12/AHSME, 19

Tags: algebra , conic , ellipse , quadratic , quadratic formula

If the line $y=mx+1$ intersects the ellipse $x^2+4y^2=1$ exactly once, then the value of $m^2$ is $\textbf{(A) }\textstyle\frac{1}{2}\qquad\textbf{(B) }\frac{2}{3}\qquad\textbf{(C) }\frac{3}{4}\qquad\textbf{(D) }\frac{4}{5}\qquad \textbf{(E) }\frac{5}{6}$

2019 AMC 12/AHSME, 6

Tags: conic , ellipse

In a given plane, points $A$ and $B$ are $10$ units apart. How many points $C$ are there in the plane such that the perimeter of $\triangle ABC$ is $50$ units and the area of $\triangle ABC$ is $100$ square units? $\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }8\qquad\textbf{(E) }\text{infinitely many}$

2008 Stars Of Mathematics, 3

Tags: conic , geometry , homothety , ellipse , incenter , parallelogram , geometric transformation

Consider a convex quadrilateral, and the incircles of the triangles determined by one of its diagonals. Prove that the tangency points of the incircles with the diagonal are symmetrical with respect to the midpoint of the diagonal if and only if the line of the incenters passes through the crossing point of the diagonals. [i]Dan Schwarz[/i]

1952 Putnam, B6

Tags: ellipse , conic , geometry

Prove the necessary and sufficient condition that a triangle inscribed in an ellipse shall have maximum area is that its centroid coincides with the center of the ellipse.

2024 Oral Moscow Geometry Olympiad, 1

Tags: geometry , ellipse , conic , bmoeg , construction

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]

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]