Found problems: 81
1999 National High School Mathematics League, 10
$P$ 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________.
2014 Belarus Team Selection Test, 1
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)
2023 Belarus Team Selection Test, 2.3
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.
1991 Baltic Way, 20
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 $OC$.
2020 Belarusian National Olympiad, 11.3
Four points $A$, $B$, $C$, $D$ lie on the hyperbola $y=\frac{1}{x}$. In triangle $BCD$ the point $A_1$ is the circumcenter of the triangle, which vertices are the midpoints of sides of $BCD$. In triangles $ACD$, $ABD$ and $ABC$ points $B_1$, $C_1$ and $D_1$ are chosen similarly. It turned out that points $A_1$, $B_1$, $C_1$ and $D_1$ are pairwise different and concyclic.
Prove that the center of that circle coincides with the $(0,0)$ point.
1956 AMC 12/AHSME, 21
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$
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.
2007 Moldova Team Selection Test, 1
Show that the plane cannot be represented as the union of the inner regions of a finite number of parabolas.
1983 IMO Longlists, 74
In a plane we are given two distinct points $A,B$ and two lines $a, b$ passing through $B$ and $A$ respectively $(a \ni B, b \ni A)$ such that the line $AB$ is equally inclined to a and b. Find the locus of points $M$ in the plane such that the product of distances from $M$ to $A$ and a equals the product of distances from $M$ to $B$ and $b$ (i.e., $MA \cdot MA' = MB \cdot MB'$, where $A'$ and $B'$ are the feet of the perpendiculars from $M$ to $a$ and $b$ respectively).
2007 Hong Kong TST, 1
[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url]
Problem 1
Let $p,q,r$ and $s$ be real numbers such that $p^{2}+q^{2}+r^{2}-s^{2}+4=0$. Find the maximum value of $3p+2q+r-4|s|$.
2020 USA TSTST, 6
Let $A$, $B$, $C$, $D$ be four points such that no three are collinear and $D$ is not the orthocenter of $ABC$. Let $P$, $Q$, $R$ be the orthocenters of $\triangle BCD$, $\triangle CAD$, $\triangle ABD$, respectively. Suppose that the lines $AP$, $BQ$, $CR$ are pairwise distinct and are concurrent. Show that the four points $A$, $B$, $C$, $D$ lie on a circle.
[i]Andrew Gu[/i]
KoMaL A Problems 2022/2023, A. 853
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]
2011 BMO TST, 3
In the acute angle triangle $ABC$ the point $O$ is the center of the circumscribed circle and the lines $OA,OB,OC$ intersect sides $BC,CA,AB$ respectively in points $M,N,P$ such that $\angle NMP=90^o$.
[b](a)[/b] Find the ratios $\frac{\angle AMN}{\angle NMC}$,$\frac{\angle AMP}{\angle PMB}$.
[b](b)[/b] If any of the angles of the triangle $ABC$ is $60^o$, find the two other angles.
2006 Pre-Preparation Course Examination, 3
There is a right angle whose vertex moves on a fixed circle and one of it's sides passes a fixed point. What is the curve that the other side of the angle is always tangent to it.
2005 China Western Mathematical Olympiad, 5
Circles $C(O_1)$ and $C(O_2)$ intersect at points $A$, $B$. $CD$ passing through point $O_1$ intersects $C(O_1)$ at point $D$ and tangents $C(O_2)$ at point $C$. $AC$ tangents $C(O_1)$ at $A$. Draw $AE \bot CD$, and $AE$ intersects $C(O_1)$ at $E$. Draw $AF \bot DE$, and $AF$ intersects $DE$ at $F$. Prove that $BD$ bisects $AF$.
2006 Belarusian National Olympiad, 5
A convex quadrilateral $ABCD$ Is placed on the Cartesian plane. Its vertices $A$ and $D$ belong to the negative branch of the graph of the hyperbola $y= 1/x$, the vertices $B$ and $C$ belong to the positive branch of the graph and point $B$ lies at the left of $C$, the segment $AC$ passes through the origin $(0,0)$. Prove that $\angle BAD = \angle BCD$.
(I, Voronovich)
PEN A Problems, 5
Let $x$ and $y$ be positive integers such that $xy$ divides $x^{2}+y^{2}+1$. Show that \[\frac{x^{2}+y^{2}+1}{xy}=3.\]
2001 Baltic Way, 9
Given a rhombus $ABCD$, find the locus of the points $P$ lying inside the rhombus and satisfying $\angle APD+\angle BPC=180^{\circ}$.
2004 Brazil National Olympiad, 1
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.
2017 AMC 10, 24
The vertices of an equilateral triangle lie on the hyperbola $xy=1,$ and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?
$\textbf{(A)} \text{ 48} \qquad \textbf{(B)} \text{ 60} \qquad \textbf{(C)} \text{ 108} \qquad \textbf{(D)} \text{ 120} \qquad \textbf{(E)} \text{ 169}$
1955 AMC 12/AHSME, 8
The graph of $ x^2\minus{}4y^2\equal{}0$:
$ \textbf{(A)}\ \text{is a hyperbola intersecting only the }x\text{ \minus{}axis} \\
\textbf{(B)}\ \text{is a hyperbola intersecting only the }y\text{ \minus{}axis} \\
\textbf{(C)}\ \text{is a hyperbola intersecting neither axis} \\
\textbf{(D)}\ \text{is a pair of straight lines} \\
\textbf{(E)}\ \text{does not exist}$
2022 Belarusian National Olympiad, 11.2
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 $m$)
1969 IMO Longlists, 5
$(BEL 5)$ 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.
2000 AIME Problems, 2
A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola $x^2-y^2=2000^2.$
1984 National High School Mathematics League, 3
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: $\text{C}_m^n=\frac{m!}{n!(m-n)!}$.
$\text{(A)}15\qquad\text{(B)}10\qquad\text{(C)}7\qquad\text{(D)}6$