Olympiad problems

2018 Belarusian National Olympiad, 11.5

The circle $S_1$ intersects the hyperbola $y=\frac1x$ at four points $A$, $B$, $C$, and $D$, and the other circle $S_2$ intersects the same hyperbola at four points $A$, $B$, $F$, and $G$. It's known that the radii of circles $S_1$ and $S_2$ are equal. Prove that the points $C$, $D$, $F$, and $G$ are the vertices of the parallelogram.

2010 USAJMO, 4

Tags: parabola , analytic geometry , nt , conic

A triangle is called a parabolic triangle if its vertices lie on a parabola $y = x^2$. Prove that for every nonnegative integer $n$, there is an odd number $m$ and a parabolic triangle with vertices at three distinct points with integer coordinates with area $(2^nm)^2$.

2011 Romanian Masters In Mathematics, 3

Tags: geometry , circumcircle , geometric transformation , trigonometry , parabola , conic

A triangle $ABC$ is inscribed in a circle $\omega$. A variable line $\ell$ chosen parallel to $BC$ meets segments $AB$, $AC$ at points $D$, $E$ respectively, and meets $\omega$ at points $K$, $L$ (where $D$ lies between $K$ and $E$). Circle $\gamma_1$ is tangent to the segments $KD$ and $BD$ and also tangent to $\omega$, while circle $\gamma_2$ is tangent to the segments $LE$ and $CE$ and also tangent to $\omega$. Determine the locus, as $\ell$ varies, of the meeting point of the common inner tangents to $\gamma_1$ and $\gamma_2$. [i](Russia) Vasily Mokin and Fedor Ivlev[/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 $ADE$.

2005 AMC 12/AHSME, 8

Tags: parabola , conic

For how many values of $ a$ is it true that the line $ y \equal{} x \plus{} a$ passes through the vertex of the parabola $ y \equal{} x^2 \plus{} a^2$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ \text{infinitely many}$

Kvant 2024, M2823

Tags: geometry , algebra , parabola , conic

A parabola $p$ is drawn on the coordinate plane — the graph of the equation $y =-x^2$, and a point $A$ is marked that does not lie on the parabola $p$. All possible parabolas $q$ of the form $y = x^2+ax+b$ are drawn through point $A$, intersecting $p$ at two points $X$ and $Y$ . Prove that all possible $XY$ lines pass through a fixed point in the plane. [i]P.A.Kozhevnikov[/i]

1987 China Team Selection Test, 1

Tags: geometry , rectangle , ratio , analytic geometry , symmetry , hyperbola , conic

Given a convex figure in the Cartesian plane that is symmetric with respect of both axis, we construct a rectangle $A$ inside it with maximum area (over all posible rectangles). Then we enlarge it with center in the center of the rectangle and ratio lamda such that is covers the convex figure. Find the smallest lamda such that it works for all convex figures.

2011 Belarus Team Selection Test, 1

Tags: parabola , circles , parallel , geometry , conic , chord

$AB$ 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

1990 Bulgaria National Olympiad, Problem 2

Tags: parabola , conic

Let be given a real number $\alpha\ne0$. Show that there is a unique point $P$ in the coordinate plane, such that for every line through $P$ which intersects the parabola $y=\alpha x^2$ in two distinct points $A$ and $B$, segments $OA$ and $OB$ are perpendicular (where $O$ is the origin).

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?

2006 Stanford Mathematics Tournament, 13

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

A ray is drawn from the origin tangent to the graph of the upper part of the hyperbola $y^2=x^2-x+1$ in the first quadrant. This ray makes an angle of $\theta$ with the positive $x$-axis. Compute $\cos\theta$.

2008 Harvard-MIT Mathematics Tournament, 31

Tags: hyperbola , analytic geometry , graphing lines , slope , conic

Let $ \mathcal{C}$ be the hyperbola $ y^2 \minus{} x^2 \equal{} 1$. Given a point $ P_0$ on the $ x$-axis, we construct a sequence of points $ (P_n)$ on the $ x$-axis in the following manner: let $ \ell_n$ be the line with slope $ 1$ passing passing through $ P_n$, then $ P_{n\plus{}1}$ is the orthogonal projection of the point of intersection of $ \ell_n$ and $ \mathcal C$ onto the $ x$-axis. (If $ P_n \equal{} 0$, then the sequence simply terminates.) Let $ N$ be the number of starting positions $ P_0$ on the $ x$-axis such that $ P_0 \equal{} P_{2008}$. Determine the remainder of $ N$ when divided by $ 2008$.

2012 Today's Calculation Of Integral, 779

Tags: calculus , integration , parabola , analytic geometry , geometry , calculus computations , conic

Consider parabolas $C_a: y=-2x^2+4ax-2a^2+a+1$ and $C: y=x^2-2x$ in the coordinate plane. When $C_a$ and $C$ have two intersection points, find the maximum area enclosed by these parabolas.

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$.

1991 National High School Mathematics League, 14

Tags: parabola , conic , geometry

$O$ is the vertex of a parabola, $F$ is its focus. $PQ$ is a chord of the parabola. If $|OF|=a,|PQ|=b$, find the area of $\triangle OPQ$.

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: $\textbf{(A)}\ \text{A straight line }AB,1\dfrac{1}{2}\text{ inches from }A \qquad\\ \textbf{(B)}\ \text{A circle with }A\text{ as center and radius }2\text{ inches} \qquad\\ \textbf{(C)}\ \text{A circle with }A\text{ as center and radius }3\text{ inches} \qquad\\ \textbf{(D)}\ \text{A circle with radius }3\text{ inches and center }4\text{ inches from }B\text{ along } BA \qquad\\ \textbf{(E)}\ \text{An ellipse with }A\text{ as focus}$

Today's calculation of integrals, 874

Tags: calculus , integration , parabola , geometry , calculus computations , conic

Given a parabola $C : y=1-x^2$ in $xy$-palne with the origin $O$. Take two points $P(p,\ 1-p^2),\ Q(q,\ 1-q^2)\ (p<q)$ on $C$. (1) Express the area $S$ of the part enclosed by two segments $OP,\ OQ$ and the parabalola $C$ in terms of $p,\ q$. (2) If $q=p+1$, then find the minimum value of $S$. (3) If $pq=-1$, then find the minimum value of $S$.

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 $\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}3\qquad\text{(D)}4$

1986 Balkan MO, 4

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

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.

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 $BC$.

1992 National High School Mathematics League, 1

Tags: parabola , conic

For any positive integer $n$, $A_n$ and $B_n$ are intersection of parabola $y=(n^2+n)x^2-(2n+1)x+1$ and $x$-axis. Then, the value of $|A_1B_1|+|A_2B_2|+\cdots+|A_{1992}B_{1992}|$ is $\text{(A)}\frac{1991}{1992}\qquad\text{(B)}\frac{1992}{1993}\qquad\text{(C)}\frac{1991}{1993}\qquad\text{(D)}\frac{1993}{1992}$

2006 China Second Round Olympiad, 9

Tags: ellipse , ratio , conic , geometry

Suppose points $F_1, F_2$ are the left and right foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{4}=1$ respectively, and point $P$ is on line $l:$, $x-\sqrt{3} y+8+2\sqrt{3}=0$. Find the value of ratio $\frac{|PF_1|}{|PF_2|}$ when $\angle F_1PF_2$ reaches its maximum value.

KoMaL A Problems 2022/2023, A. 853

Tags: euler circle , hyperbola , simson line , geometry , conic

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]

2022 IMO Shortlist, G6

Tags: hyperbola , geometry , projective geometry , conic

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.