Olympiad problems

2011 Belarus Team Selection Test, 1

Tags: conics , parabola , circles , Chords , parallel , geometry

$AB$ and $CD$ are two parallel chords of a parabola. Circle $S_1$ passing through points $A,B$ intersects circle $S_2$ passing through $C,D$ at points $E,F$. Prove that if $E$ belongs to the parabola, then $F$ also belongs to the parabola. I.Voronovich

1999 Federal Competition For Advanced Students, Part 2, 2

Tags: geometry , 3D geometry , sphere , geometric transformation , conics , ellipse , geometry unsolved

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.

2021 China Second Round Olympiad, Problem 3

Tags: conics , ellipse , complex numbers

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]

2004 Germany Team Selection Test, 3

Tags: geometry , analytic geometry , circumcircle , incenter , trigonometry , ratio , conics

Given six real numbers $a$, $b$, $c$, $x$, $y$, $z$ such that $0 < b-c < a < b+c$ and $ax + by + cz = 0$. What is the sign of the sum $ayz + bzx + cxy$ ?

1969 IMO Shortlist, 1

Tags: conics , parabola , geometry , Locus , Locus problems , IMO Shortlist , IMO Longlist

$(BEL 1)$ 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 $P_1.$

2002 National High School Mathematics League, 13

Tags: conics , parabola

$A(0,2)$, and two points $B,C$ on parabola $y^2=x+4$ satisfy that $AB\perp BC$. Find the range value of $y_C$.

2002 Czech and Slovak Olympiad III A, 5

Tags: geometry , rectangle , analytic geometry , trigonometry , parameterization , conics , graphing lines

A triangle $KLM$ is given in the plane together with a point $A$ lying on the half-line opposite to $KL$. Construct a rectangle $ABCD$ whose vertices $B, C$ and $D$ lie on the lines $KM, KL$ and $LM$, respectively. (We allow the rectangle to be a square.)

2024 HMIC, 5

Tags: conics , hyperbola , geometry , HMIC

Let $ABC$ be an acute, scalene triangle with circumcenter $O$ and symmedian point $K$. Let $X$ be the point on the circumcircle of triangle $BOC$ such that $\angle AXO = 90^\circ$. Assume that $X\neq K$. The hyperbola passing through $B$, $C$, $O$, $K$, and $X$ intersects the circumcircle of triangle $ABC$ at points $U$ and $V$, distinct from $B$ and $C$. Prove that $UV$ is the perpendicular bisector of $AX$. [i]The symmedian point of triangle $ABC$ is the intersection of the reflections of $B$-median and $C$-median across the angle bisectors of $\angle ABC$ and $\angle ACB$, respectively.[/i] [i]Pitchayut Saengrungkongka[/i]

1952 Miklós Schweitzer, 2

Tags: conics , geometry proposed , geometry

Is it possible to find three conics in the plane such that any straight line in the plane intersects at least two of the conics and through any point of the plane pass tangents to at least two of them?

2010 Purple Comet Problems, 26

Tags: analytic geometry , conics , parabola , symmetry , graphing lines , slope , number theory

In the coordinate plane a parabola passes through the points $(7,6)$, $(7,12)$, $(18,19)$, and $(18,48)$. The axis of symmetry of the parabola is a line with slope $\tfrac{r}{s}$ where r and s are relatively prime positive integers. Find $r + s$.

1992 AMC 12/AHSME, 14

Tags: conics , AMC

Which of the following equations have the same graph? $I.\ y = x - 2$ $II.\ y = \frac{x^{2} - 4}{x + 2}$ $III.\ (x + 2)y = x^{2} - 4$ $ \textbf{(A)}\ \text{I and II only} $ $ \textbf{(B)}\ \text{I and III only} $ $ \textbf{(C)}\ \text{II and III only} $ $ \textbf{(D)}\ \text{I, II and III} $ $ \textbf{(E)}\ \text{None. All equations have different graphs} $

1964 AMC 12/AHSME, 25

Tags: algebra , polynomial , conics , parameterization , AMC

The set of values of $m$ for which $x^2+3xy+x+my-m$ has two factors, with integer coefficients, which are linear in $x$ and $y$, is precisely: $ \textbf{(A)}\ 0, 12, -12\qquad\textbf{(B)}\ 0, 12\qquad\textbf{(C)}\ 12, -12\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 0 $

2023 Belarus Team Selection Test, 2.3

Tags: conics , hyperbola , geometry , projective geometry

Let $ABC$ be an acute triangle with altitude $\overline{AH}$, and let $P$ be a variable point such that the angle bisectors $k$ and $\ell$ of $\angle PBC$ and $\angle PCB$, respectively, meet on $\overline{AH}$. Let $k$ meet $\overline{AC}$ at $E$, $\ell$ meet $\overline{AB}$ at $F$, and $\overline{EF}$ meet $\overline{AH}$ at $Q$. Prove that as $P$ varies, line $PQ$ passes through a fixed point.

2011 China Second Round Olympiad, 11

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

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]

2024 China Team Selection Test, 2

Tags: Kiepert Hyperbola , geometry , conics , 2024 CTST

In acute triangle $\triangle {ABC}$, $\angle A > \angle B > \angle C$. $\triangle {AC_1B}$ and $\triangle {CB_1A}$ are isosceles triangles such that $\triangle {AC_1B} \stackrel{+}{\sim} \triangle {CB_1A}$. Let lines $BB_1, CC_1$ intersects at ${T}$. Prove that if all points mentioned above are distinct, $\angle ATC$ isn't a right angle.

2002 Romania National Olympiad, 1

Tags: conics , hyperbola , analytic geometry , geometry proposed , geometry

In the Cartesian plane consider the hyperbola \[\Gamma=\{M(x,y)\in\mathbb{R}^2 \vert \frac{x^2}{4}-y^2=1\} \] and a conic $\Gamma '$, disjoint from $\Gamma$. Let $n(\Gamma ,\Gamma ')$ be the maximal number of pairs of points $(A,A')\in\Gamma\times\Gamma '$ such that $AA'\le BB'$, for any $(B,B')$ For each $p\in\{0,1,2,4\}$, find the equation of $\Gamma'$ for which $n(\Gamma ,\Gamma ')=p$. Justify the answer.

2003 AMC 12-AHSME, 25

Tags: algebra , function , domain , conics , parabola , calculus , inequalities

Let $ f(x)\equal{}\sqrt{ax^2\plus{}bx}$. For how many real values of $ a$ is there at least one positive value of $ b$ for which the domain of $ f$ and the range of $ f$ are the same set? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \text{infinitely many}$

1996 IMC, 10

Tags: conics , ellipse , geometry

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 $l(A)\geq \frac{\pi}{2}$.

1988 National High School Mathematics League, 2

Tags: conics , ellipse

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 $\text{(A)}|k|>1\qquad\text{(B)}|k|\neq1\qquad\text{(C)}-1<k<1\qquad\text{(D)}0<|k|<1$

2005 International Zhautykov Olympiad, 2

Tags: geometry , perimeter , conics , analytic geometry , linear algebra , parallelogram

Let the circle $ (I; r)$ be inscribed in the triangle $ ABC$. Let $ D$ be the point of contact of this circle with $ BC$. Let $ E$ and $ F$ be the midpoints of $ BC$ and $ AD$, respectively. Prove that the three points $ I$, $ E$, $ F$ are collinear.

1997 National High School Mathematics League, 4

Tags: conics , ellipse

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 $\text{(A)}(0,1)\qquad\text{(B)}(1,+\infty)\qquad\text{(C)}(0,5)\qquad\text{(D)}(5,+\infty)$

2019 CMIMC, 3

Tags: 2019 , team , conics , parabola

Points $A(0,0)$ and $B(1,1)$ are located on the parabola $y=x^2$. A third point $C$ is positioned on this parabola between $A$ and $B$ such that $AC=CB=r$. What is $r^2$?

2000 National High School Mathematics League, 10

Tags: conics , 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 $\angle ABF=$________.

1980 Putnam, A1

Tags: Putnam , conics , parabola , algebra , polynomial , function

Let $b$ and $c$ be fixed real numbers and let the ten points $(j,y_j )$ for $j=1,2,\ldots,10$ lie on the parabola $y =x^2 +bx+c.$ For $j=1,2,\ldots, 9$ let $I_j$ be the intersection of the tangents to the given parabola at $(j, y_j )$ and $(j+1, y_{j+1}).$ Determine the poynomial function $y=g(x)$ of least degree whose graph passes through all nine points $I_j .$

1991 Arnold's Trivium, 36

Tags: conics , parabola

Sketch the evolvent of the cubic parabola $y=x^3$ (the evolvent is the locus of the points $\overrightarrow{r}(s)+(c-s)\dot{\overrightarrow{r}}(s)$, where $s$ is the arc-length of the curve $\overrightarrow{r}(s)$ and $c$ is a constant).