Found problems: 81
1998 USAMO, 6
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 $A_{n+j} = A_{j}$.)
2008 Hungary-Israel Binational, 3
P and Q are 2 points in the area bounded by 2 rays, e and f, coming out from a point O. Describe how to construct, with a ruler and a compass only, an isosceles triangle ABC, such that his base AB is on the ray e, the point C is on the ray f, P is on AC, and Q on BC.
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.
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.
Today's calculation of integrals, 891
Given a triangle $OAB$ with the vetices $O(0,\ 0,\ 0),\ A(1,\ 0,\ 0),\ B(1,\ 1,\ 0)$ in the $xyz$ space.
Let $V$ be the cone obtained by rotating the triangle around the $x$-axis.
Find the volume of the solid obtained by rotating the cone $V$ around the $y$-axis.
2009 Today's Calculation Of Integral, 494
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]
1969 IMO Shortlist, 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.
1941 Putnam, B4
Given two perpendicular diameters $AB$ and $CD$ of an ellipse, we say that the diameter $A'B'$ is conjugate to $AB$ if $A'B'$ is parallel to the tangent to the ellipse at $A$. Let $A'B'$ be conjugate to $AB$ and $C'D'$ be conjugate to $CD$.
Prove that the rectangular hyperbola through $A', B', C'$ and $D'$ passes through the foci of the ellipse.
1990 Vietnam National Olympiad, 1
A triangle $ ABC$ is given in the plane. Let $ M$ be a point inside the triangle and $ A'$, $ B'$, $ C'$ be its projections on the sides $ BC$, $ CA$, $ AB$, respectively. Find the locus of $ M$ for which $ MA \cdot MA' \equal{} MB \cdot MB' \equal{} MC \cdot MC'$.
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 $3p+2q+r-4|s|$.
2007 Putnam, 2
Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $ xy\equal{}1$ and both branches of the hyperbola $ xy\equal{}\minus{}1.$ (A set $ S$ in the plane is called [i]convex[/i] if for any two points in $ S$ the line segment connecting them is contained in $ S.$)
1990 National High School Mathematics League, 3
Left focal point and right focal point of a hyperbola are $F_1,F_2$, left focal point and right focal point of a hyperbola are $M,N$. If $P$ is a point on the hyperbola, then the tangent point of inscribed circle of $\triangle PF_1F_2$ on $F_1F_2$ is
$\text{(A)}$a point on segment $MN$
$\text{(B)}$a point on segment $F_1M$ or $F_2N$
$\text{(C)}$point $M$ or $N$
$\text{(D)}$not sure
2013 Waseda University Entrance Examination, 1
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 $l_1,\ l_2$.
1984 AMC 12/AHSME, 22
Let $a$ and $c$ be fixed positive numbers. For each real number $t$ let $(x_t, y_t)$ be the vertex of the parabola $y = ax^2+bx+c$. If the set of vertices $(x_t, y_t)$ for all real values of $t$ is graphed in the plane, the graph is
A. a straight line
B. a parabola
C. part, but not all, of a parabola
D. one branch of a hyperbola
E. None of these
1997 National High School Mathematics League, 8
Line $l$ that passes right focal point of hyperbola $x^2-\frac{y^2}{2}=1$ intersects the hyperbola at $A,B$. The number of line $l$ that $|AB|=\lambda$ is 3, then $\lambda=$________.
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.
1941 Putnam, B1
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.
2007 Today's Calculation Of Integral, 211
When the parabola which has the axis parallel to $y$ -axis and passes through the origin touch to the rectangular hyperbola $xy=1$ in the first quadrant moves,
prove that the area of the figure sorrounded by the parabola and the $x$-axis is constant.
2019 Belarusian National Olympiad, 11.1
[b]a)[/b] Find all real numbers $a$ such that the parabola $y=x^2-a$ and the hyperbola $y=1/x$ intersect each other in three different points.
[b]b)[/b] Find the locus of centers of circumcircles of such triples of intersection points when $a$ takes all possible values.
[i](I. Gorodnin)[/i]
2008 Moldova National Olympiad, 12.7
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$.
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]
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}$
PEN A Problems, 5
Let $x$ and $y$ be positive integers such that $xy$ divides $x^{2}+y^{2}+1$. Show that \[\frac{x^{2}+y^{2}+1}{xy}=3.\]
2008 Harvard-MIT Mathematics Tournament, 31
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$.
2023 Brazil Team Selection Test, 4
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.