Found problems: 81
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.
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.
See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url]
Triangle $ABC$ and straight line $l$ are given at the plane. Construct using a compass and a ruler the straightline which is parallel to $l$ and bisects the area of triangle $ABC.$
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: $\text{C}_m^n=\frac{m!}{n!(m-n)!}$.
$\text{(A)}15\qquad\text{(B)}10\qquad\text{(C)}7\qquad\text{(D)}6$
The circle of center $I$ is inscribed in the convex quadrilateral $ABCD$. Let $M$ and $N$ be points on the segments $AI$ and $CI$, respectively, such that $\angle MBN = \frac 12 \angle ABC$. Prove that $\angle MDN = \frac 12 \angle ADC$.
Given a rhombus $ABCD$, find the locus of the points $P$ lying inside the rhombus and satisfying $\angle APD+\angle BPC=180^{\circ}$.
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.
$P$ 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________.
Let $n \geq 5$ be an integer. Find the largest integer $k$ (as a function of $n$) such that there exists a convex $n$-gon $A_{1}A_{2}\dots A_{n}$ for which exactly $k$ of the quadrilaterals $A_{i}A_{i+1}A_{i+2}A_{i+3}$ have an inscribed circle. (Here $A_{n+j} = A_{j}$.)
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$.
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.
Suppose the curve $ C: y \equal{} ax^3 \plus{} 4x\ (a\neq 0)$ has a common tangent line at the point $ P$ with the hyperbola $ xy \equal{} 1$ in the first quadrant.
(1) Find the value of $ a$ and the coordinate of the point $ P$.
(2) Find the volume formed by the revolution of the solid of the figure bounded by the line segment $ OP$ and the curve $ C$ about the line $ OP$.
[color=green][Edited.][/color]
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?
$\textbf{(A)} \text{ 48} \qquad \textbf{(B)} \text{ 60} \qquad \textbf{(C)} \text{ 108} \qquad \textbf{(D)} \text{ 120} \qquad \textbf{(E)} \text{ 169}$
The positive reals $a,b,c,d$ satisfy $abcd=1$. Prove that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{9}{{a + b + c + d}} \geqslant \frac{{25}}{4}$.
The points of intersection of $ xy \equal{} 12$ and $ x^2 \plus{} y^2 \equal{} 25$ are joined in succession. The resulting figure is:
$ \textbf{(A)}\ \text{a straight line} \qquad\textbf{(B)}\ \text{an equilateral triangle} \qquad\textbf{(C)}\ \text{a parallelogram}$
$ \textbf{(D)}\ \text{a rectangle} \qquad\textbf{(E)}\ \text{a square}$
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.
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.
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]
A particle $(x,y)$ moves so that its angular velocities about $(1,0)$ and $(-1,0)$ are equal in magnitude but opposite in sign. Prove that
$$y(x^2 +y^2 +1)\; dx= x(x^2 +y^2 -1) \;dy,$$
and verify that this is the differential equation of the family of rectangular hyperbolas passing through $(1,0)$ and $(-1,0)$ and having the origin as center.
Given a parabola $C: y^2=4px\ (p>0)$ with focus $F(p,\ 0)$. Let two lines $l_1,\ l_2$ passing through $F$ intersect orthogonaly each other,
$C$ intersects with $l_1$ at two points $P_1,\ P_2$ and $C$ intersects with $l_2$ at two points $Q_1,\ Q_2$. Answer the following questions.
(1) Set the equation of $l_1$ as $x=ay+p$ and let the coordinates of $P_1,\ P_2$ as $(x_1,\ y_1),\ (x_2,\ y_2)$, respectively. Express $y_1+y_2,\ y_1y_2$ in terms of $a,\ p$.
(2) Show that $\frac{1}{P_1P_2}+\frac{1}{Q_1Q_2}$ is constant regardless of way of taking $l_1,\ l_2$.
In triangle $ABC$ let the $A$-foot be $E$ and the $B$-excenter be $L$. Suppose the incircle of $ABC$ is tangent to $AC$ at $I$. Construct a hyperbola $\mathcal H$ through $A$ with $B$ and $C$ as the foci such that $A$ lies on the branch of the $\mathcal H$ closer to $C$. Construct an ellipse $\mathcal E$ passing through $I$ with $B$ and $C$ as the foci. Suppose $\mathcal E$ meets $\overline{AB}$ again at point $H$. Let $\overline{CH}$ and $\overline{BI}$ intersect the $C$-branch of $\mathcal H$ at points $M$ and $O$ respectively. Prove $E$, $L$, $M$, $O$ are concyclic.
Proposed by [i]Alex Wang[/i]
$A(-1,1)$, $B,C$ are points on hyperbola $x^2-y^2=1$. If $\triangle ABC$ is a regular triangle, then the area of $\triangle ABC$ is
$\text{(A)}\frac{\sqrt3}{3}\qquad\text{(B)}\frac{3\sqrt3}{2}\qquad\text{(C)}3\sqrt3\qquad\text{(D)}6\sqrt3\qquad$
Show that the distance between a point on the hyperbola $xy=5$ and a point on the ellipse $x^{2}+6y^{2}=6$ is at least $\frac{9}{7}$.
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]
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 $OC$.