Olympiad problems

2013 Today's Calculation Of Integral, 870

Tags: calculus , integration , ellipse , hyperbola , geometry , inequalities , conic

Consider the ellipse $E: 3x^2+y^2=3$ and the hyperbola $H: xy=\frac 34.$ (1) Find all points of intersection of $E$ and $H$. (2) Find the area of the region expressed by the system of inequality \[\left\{ \begin{array}{ll} 3x^2+y^2\leq 3 &\quad \\ xy\geq \frac 34 , &\quad \end{array} \right.\]

1991 AMC 12/AHSME, 18

Tags: hyperbola , parabola , complex number , conic

If $S$ is the set of points $z$ in the complex plane such that $(3+4i)z$ is a real number, then $S$ is a $ \textbf{(A)}\text{ right triangle}\qquad\textbf{(B)}\text{ circle}\qquad\textbf{(C)}\text{ hyperbola}\qquad\textbf{(D)}\text{ line}\qquad\textbf{(E)}\text{ parabola} $

1991 Baltic Way, 20

Tags: geometry , hyperbola , function , conic

Consider two points $A(x_1, y_1)$ and $B(x_2, y_2)$ on the graph of the function $y = \frac{1}{x}$ such that $0 < x_1 < x_2$ and $AB = 2 \cdot OA$, where $O = (0, 0)$. Let $C$ be the midpoint of the segment $AB$. Prove that the angle between the $x$-axis and the ray $OA$ is equal to three times the angle between the $x$-axis and the ray $OC$.

2004 Brazil National Olympiad, 1

Tags: geometry , rhombus , hyperbola , symmetry , inradius , incenter , conic

Let $ABCD$ be a convex quadrilateral. Prove that the incircles of the triangles $ABC$, $BCD$, $CDA$ and $DAB$ have a point in common if, and only if, $ABCD$ is a rhombus.

1969 IMO Longlists, 5

Tags: hyperbola , geometry , circumcircle , conic

$(BEL 5)$ Let $G$ be the centroid of the triangle $OAB.$ $(a)$ Prove that all conics passing through the points $O,A,B,G$ are hyperbolas. $(b)$ Find the locus of the centers of these hyperbolas.

1969 IMO Shortlist, 2

Tags: hyperbola , geometry , circumcircle , parallelogram , conic

$(BEL 2) (a)$ Find the equations of regular hyperbolas passing through the points $A(\alpha, 0), B(\beta, 0),$ and $C(0, \gamma).$ $(b)$ Prove that all such hyperbolas pass through the orthocenter $H$ of the triangle $ABC.$ $(c)$ Find the locus of the centers of these hyperbolas. $(d)$ Check whether this locus coincides with the nine-point circle of the triangle $ABC.$

1999 National High School Mathematics League, 10

Tags: hyperbola , conic

$P$ is a point on hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$, if the distance from $P$ to right directrix is the arithmetic mean of the distance from $P$ to two focal points, then the $x$-axis of $P$ is________.

1995 VJIMC, Problem 1

Tags: hyperbola , conic

Prove that the systems of hyperbolas \begin{align*}x^2-y^2&=a\\xy&=b\end{align*}are orthogonal.

2005 IberoAmerican Olympiad For University Students, 4

Tags: hyperbola , function , asymptote , analytic geometry , trigonometry , geometry , conic

A variable tangent $t$ to the circle $C_1$, of radius $r_1$, intersects the circle $C_2$, of radius $r_2$ in $A$ and $B$. The tangents to $C_2$ through $A$ and $B$ intersect in $P$. Find, as a function of $r_1$ and $r_2$, the distance between the centers of $C_1$ and $C_2$ such that the locus of $P$ when $t$ varies is contained in an equilateral hyperbola. [b]Note[/b]: A hyperbola is said to be [i]equilateral[/i] if its asymptotes are perpendicular.

2023 Indonesia TST, 3

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.

2008 Moldova MO 11-12, 7

Tags: hyperbola , geometry , conic

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$.

1999 Federal Competition For Advanced Students, Part 2, 3

Tags: hyperbola , parabola , function , probability , floor function , conic

Find all pairs $(x, y)$ of real numbers such that \[y^2 - [x]^2 = 19.99 \text{ and } x^2 + [y]^2 = 1999\] where $f(x)=[x]$ is the floor function.

2000 National High School Mathematics League, 3

Tags: hyperbola , geometry , conic

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

2022 Belarusian National Olympiad, 11.2

Tags: hyperbola , analytic geometry , algebra , conic

Two perpendicular lines pass through the point $F(1;1)$ of coordinate plane. One of them intersects hyperbola $y=\frac{1}{2x}$ at $A$ and $C$ ($C_x>A_x$), and the other one intersects the left part of hyperbola at $B$ and the right at $D$. Let $m=(C_x-A_x)(D_x-B_x)$ Find the area of non-convex quadraliteral $ABCD$ (in terms of $m$)

2010 Putnam, B2

Tags: analytic geometry , inequalities , geometry , rectangle , hyperbola , conic

Given that $A,B,$ and $C$ are noncollinear points in the plane with integer coordinates such that the distances $AB,AC,$ and $BC$ are integers, what is the smallest possible value of $AB?$

KoMaL A Problems 2022/2023, A. 853

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

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]

1984 National High School Mathematics League, 3

Tags: analytic geometry , hyperbola , conic

For any integers $1\leq n\leq m\leq5$, how many hyperbolas does the equation $\rho=\frac{1}{1-\text{C}_m^n \cos\theta}$ represent? Note: $\text{C}_m^n=\frac{m!}{n!(m-n)!}$. $\text{(A)}15\qquad\text{(B)}10\qquad\text{(C)}7\qquad\text{(D)}6$

2014 Belarus Team Selection Test, 1

Tags: geometry , analytic geometry , hyperbola , parallel , parabola , conic

All vertices of triangles $ABC$ and $A_1B_1C_1$ lie on the hyperbola $y=1/x$. It is known that $AB \parallel A_1B_1$ and $BC \parallel B_1C_1$. Prove that $AC_1 \parallel A_1C$. (I. Gorodnin)

1956 AMC 12/AHSME, 21

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

If each of two intersecting lines intersects a hyperbola and neither line is tangent to the hyperbola, then the possible number of points of intersection with the hyperbola is: $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 2\text{ or }3 \qquad\textbf{(C)}\ 2\text{ or }4 \qquad\textbf{(D)}\ 3\text{ or }4 \qquad\textbf{(E)}\ 2,3,\text{ or }4$

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.

2022 IMO Shortlist, G6