Olympiad problems

2023 Indonesia TST, 3

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.

2006 China Second Round Olympiad, 13

Tags: conics , parabola

Given an integer $n\ge 2$, define $M_0 (x_0, y_0)$ to be an intersection point of the parabola $y^2=nx-1$ and the line $y=x$. Prove that for any positive integer $m$, there exists an integer $k\ge 2$ such that $(x^m_0, y^m_0)$ is an intersection point of $y^2=mx-1$ and the line $y=x$.

1997 National High School Mathematics League, 14

Tags: conics , hyperbola

Two branches of the hyperbola $xy=1$ are $C_1,C_2$ ($C_1$ in Quadrant I, $C_2$ in Quadrant III). Three apexes of regular triangle $PQR$ are on the hyperbola. [b](a)[/b] $P,Q,R$ cannot be on the same branch. [b](b)[/b] $P(-1,-1)$ is a point on $C_2$, if $Q,R$ are on $C_1$, find their coordinates.

2012 Math Prize For Girls Problems, 16

Tags: geometry , conics , ellipse , analytic geometry , absolute value

Say that a complex number $z$ is [i]three-presentable[/i] if there is a complex number $w$ of absolute value $3$ such that $z = w - \frac{1}{w}$. Let $T$ be the set of all three-presentable complex numbers. The set $T$ forms a closed curve in the complex plane. What is the area inside $T$?

1965 AMC 12/AHSME, 9

Tags: conics , parabola , calculus

The vertex of the parabola $ y \equal{} x^2 \minus{} 8x \plus{} c$ will be a point on the $ x$-axis if the value of $ c$ is: $ \textbf{(A)}\ \minus{} 16 \qquad \textbf{(B)}\ \minus{} 4 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 16$

1997 Spain Mathematical Olympiad, 3

Tags: conics , parabola , geometry , circumcircle , Triangle

For each parabola $y = x^2+ px+q$ intersecting the coordinate axes in three distinct points, consider the circle passing through these points. Prove that all these circles pass through a single point, and find this point.

1962 AMC 12/AHSME, 26

Tags: conics , parabola , calculus

For any real value of $ x$ the maximum value of $ 8x \minus{} 3x^2$ is: $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ \frac83 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ \frac{16}{3}$

1973 Polish MO Finals, 6

Tags: conics , ellipse , geometry , Geometric Inequalities

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

1993 All-Russian Olympiad Regional Round, 11.5

Tags: conics , parabola , combinatorics unsolved , combinatorics

The expression $ x^3 \plus{} . . . x^2 \plus{} . . . x \plus{} ... \equal{} 0$ is written on the blackboard. Two pupils alternately replace the dots by real numbers. The first pupil attempts to obtain an equation having exactly one real root. Can his opponent spoil his efforts?

2020 JHMT, 3

Tags: geometry , conics , ellipse

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.

1986 Balkan MO, 4

Tags: geometry , perimeter , conics , hyperbola , parallelogram , rectangle , perpendicular bisector

Let $ABC$ a triangle and $P$ a point such that the triangles $PAB, PBC, PCA$ have the same area and the same perimeter. Prove that if: a) $P$ is in the interior of the triangle $ABC$ then $ABC$ is equilateral. b) $P$ is in the exterior of the triangle $ABC$ then $ABC$ is right angled triangle.

2005 Harvard-MIT Mathematics Tournament, 7

Tags: calculus , conics , parabola

Two ants, one starting at $ (-1, 1) $, the other at $ (1, 1) $, walk to the right along the parabola $ y = x^2 $ such that their midpoint moves along the line $ y = 1 $ with constant speed $1$. When the left ant first hits the line $ y = \frac {1}{2} $, what is its speed?

2015 AMC 10, 24

Tags: geometry , perimeter , calculus , rhombus , conics , parabola

For some positive integers $p$, there is a quadrilateral $ABCD$ with positive integer side lengths, perimeter $p$, right angles at $B$ and $C$, $AB=2$, and $CD=AD$. How many different values of $p<2015$ are possible? $\textbf{(A) }30\qquad\textbf{(B) }31\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$

2016 Israel Team Selection Test, 3

Tags: geometry , conics , 3D geometry , octahedron

Prove that there exists an ellipsoid touching all edges of an octahedron if and only if the octahedron's diagonals intersect. (Here an octahedron is a polyhedron consisting of eight triangular faces, twelve edges, and six vertices such that four faces meat at each vertex. The diagonals of an octahedron are the lines connecting pairs of vertices not connected by an edge).

2000 National High School Mathematics League, 3

Tags: conics , hyperbola , geometry

$A(-1,1)$, $B,C$ are points on hyperbola $x^2-y^2=1$. If $\triangle ABC$ is a regular triangle, then the area of $\triangle ABC$ is $\text{(A)}\frac{\sqrt3}{3}\qquad\text{(B)}\frac{3\sqrt3}{2}\qquad\text{(C)}3\sqrt3\qquad\text{(D)}6\sqrt3\qquad$

2016 ASMT, T3

Tags: conics , ellipse , geometry , asmt , analytic geometry

An ellipse lies in the $xy$-plane and is tangent to both the $x$-axis and $y$-axis. Given that one of the foci is at $(9, 12)$, compute the minimum possible distance between the two foci.

2008 Harvard-MIT Mathematics Tournament, 3

Tags: quadratics , inequalities , function , conics , parabola , analytic geometry , absolute value

Determine all real numbers $ a$ such that the inequality $ |x^2 \plus{} 2ax \plus{} 3a|\le2$ has exactly one solution in $ x$.

2002 Iran MO (3rd Round), 15

Tags: ratio , conics , function , geometry proposed , geometry

Let A be be a point outside the circle C, and AB and AC be the two tangents from A to this circle C. Let L be an arbitrary tangent to C that cuts AB and AC in P and Q. A line through P parallel to AC cuts BC in R. Prove that while L varies, QR passes through a fixed point. :)

2009 Today's Calculation Of Integral, 521

Tags: calculus , integration , geometry , conics , parabola , vector , function

Let $ t$ be a positive number. Draw two tangent lines from the point $ (t, \minus{} 1)$ to the parabpla $ y \equal{} x^2$. Denote $ S(t)$ the area bounded by the tangents line and the parabola. Find the minimum value of $ \frac {S(t)}{\sqrt {t}}$.

2008 Moldova National Olympiad, 12.7

Tags: geometry , conics , hyperbola , geometry proposed

Triangle $ ABC$ has fixed vertices $ B$ and $ C$, so that $ BC \equal{} 2$ and $ A$ is variable. Denote by $ H$ and $ G$ the orthocenter and the centroid, respectively, of triangle $ ABC$. Let $ F\in(HG)$ so that $ \frac {HF}{FG} \equal{} 3$. Find the locus of the point $ A$ so that $ F\in BC$.

2008 IberoAmerican Olympiad For University Students, 4

Tags: conics , parabola , geometry , geometry proposed

Two vertices $A,B$ of a triangle $ABC$ are located on a parabola $y=ax^2 + bx + c$ with $a>0$ in such a way that the sides $AC,BC$ are tangent to the parabola. Let $m_c$ be the length of the median $CC_1$ of triangle $ABC$ and $S$ be the area of triangle $ABC$. Find \[\frac{S^2}{m_c^3}\]

1959 AMC 12/AHSME, 8

Tags: quadratics , conics , parabola , analytic geometry , AMC , optimizing quadratics , algebra

The value of $x^2-6x+13$ can never be less than: $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 4.5 \qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 13 $

2014 Turkey Team Selection Test, 2

Tags: geometry , circumcircle , geometric transformation , homothety , trigonometry , conics , inequalities

A circle $\omega$ cuts the sides $BC,CA,AB$ of the triangle $ABC$ at $A_1$ and $A_2$; $B_1$ and $B_2$; $C_1$ and $C_2$, respectively. Let $P$ be the center of $\omega$. $A'$ is the circumcenter of the triangle $A_1A_2P$, $B'$ is the circumcenter of the triangle $B_1B_2P$, $C'$ is the circumcenter of the triangle $C_1C_2P$. Prove that $AA', BB'$ and $CC'$ concur.

KoMaL A Problems 2022/2023, A. 853

Tags: conics , hyperbola , Simson line , Nine-point circle , geometry , komal

Let points $A, B, C, A', B', C'$ be chosen in the plane such that no three of them are collinear, and let lines $AA'$, $BB'$ and $CC'$ be tangent to a given equilateral hyperbola at points $A$, $B$ and $C$, respectively. Assume that the circumcircle of $A'B'C'$ is the same as the nine-point circle of triangle $ABC$. Let $s(A')$ be the Simson line of point $A'$ with respect to the orthic triangle of $ABC$. Let $A^*$ be the intersection of line $B'C'$ and the perpendicular on $s(A')$ from the point $A$. Points $B^*$ and $C^*$ are defined in a similar manner. Prove that points $A^*$, $B^*$ and $C^*$ are collinear. [i]Submitted by Áron Bán-Szabó, Budapest[/i]

2010 Sharygin Geometry Olympiad, 22

Tags: conics , parabola , geometry , geometric transformation , reflection , trapezoid , symmetry

A circle centered at a point $F$ and a parabola with focus $F$ have two common points. Prove that there exist four points $A, B, C, D$ on the circle such that the lines $AB, BC, CD$ and $DA$ touch the parabola.