Olympiad problems

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.

2007 Vietnam National Olympiad, 3

Tags: analytic geometry , hyperbola , geometry , conic

Let B,C be fixed points and A be roving point. Let H, G be orthecentre and centroid of triagle ABC. Known midpoint of HG lies on BC, find locus of A

2002 Romania National Olympiad, 1

Tags: hyperbola , analytic geometry , geometry , conic

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.

1969 IMO Longlists, 2

Tags: hyperbola , circumcircle , parallelogram , conic , geometry

$(BEL 2) (a)$ 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 $ABC.$

2011 BMO TST, 3

Tags: ratio , hyperbola , trigonometry , analytic geometry , circumcircle , geometry , conic

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.

2005 National High School Mathematics League, 5

Tags: ellipse , hyperbola , conic

Which kind of curve does the equation $\frac{x^2}{\sin\sqrt2-\sin\sqrt3}+\frac{y^2}{\cos\sqrt2-\cos\sqrt3}=1$ refer to? $\text{(A)}$ An ellipse, whose focal points are on $x$-axis. $\text{(B)}$ A hyperbola, whose focal points are on $x$-axis. $\text{(C)}$ An ellipse, whose focal points are on $y$-axis. $\text{(D)}$ A hyperbola, whose focal points are on $y$-axis.