Olympiad problems

2019 Argentina National Olympiad Level 2, 3

Let $\Gamma$ be a circle of center $S$ and radius $r$ and let be $A$ a point outside the circle. Let $BC$ be a diameter of $\Gamma$ such that $B$ does not belong to the line $AS$ and consider the point $O$ where the perpendicular bisectors of triangle $ABC$ intersect, that is, the circumcenter of $ABC$. Determine all possible locations of point $O$ when $B$ varies in circle $\Gamma$.

2016 Chile National Olympiad, 6

Tags: geometry , Locus , 3D geometry , Locus problems , collinear

Let $P_1$ and $P_2$ be two non-parallel planes in space, and $A$ a point that does not It is in none of them. For each point $X$, let $X_1$ denote its reflection with respect to $P_1$, and $X_2$ its reflection with respect to $P_2$. Determine the locus of points $X$ for the which $X_1, X_2$ and $A$ are collinear.

2006 Sharygin Geometry Olympiad, 24

Tags: geometry , Locus , 3D geometry , projections , Plane , circle , sphere

a) Two perpendicular rays are drawn through a fixed point $P$ inside a given circle, intersecting the circle at points $A$ and $B$. Find the geometric locus of the projections of $P$ on the lines $AB$. b) Three pairwise perpendicular rays passing through the fixed point $P$ inside a given sphere intersect the sphere at points $A, B, C$. Find the geometrical locus of the projections $P$ on the $ABC$ plane

1969 IMO Longlists, 1

Tags: conics , parabola , geometry , Locus , Locus problems , IMO Shortlist , IMO Longlist

$(BEL 1)$ A parabola $P_1$ with equation $x^2 - 2py = 0$ and parabola $P_2$ with equation $x^2 + 2py = 0, p > 0$, are given. A line $t$ is tangent to $P_2.$ Find the locus of pole $M$ of the line $t$ with respect to $P_1.$

1966 IMO Longlists, 28

Tags: geometry , circle , Locus , Locus problems , IMO Longlist , IMO Shortlist

In the plane, consider a circle with center $S$ and radius $1.$ Let $ABC$ be an arbitrary triangle having this circle as its incircle, and assume that $SA\leq SB\leq SC.$ Find the locus of [b]a.)[/b] all vertices $A$ of such triangles; [b]b.)[/b] all vertices $B$ of such triangles; [b]c.)[/b] all vertices $C$ of such triangles.

1969 IMO Shortlist, 39

Tags: geometry , 3D geometry , cube , Locus , Locus problems , IMO Shortlist , IMO Longlist

$(HUN 6)$ Find the positions of three points $A,B,C$ on the boundary of a unit cube such that $min\{AB,AC,BC\}$ is the greatest possible.

1987 Traian Lălescu, 2.1

Tags: geometry , rectangle , Locus

Let $ ABCD $ be a rectangle that has $ M $ on its $ BD $ diagonal. If $ N,P $ are the projections of $ M $ on $ AB, $ respectively, $ AD, $ what's the locus of the intersection between $ CP $ and $ DN? $

1983 Bundeswettbewerb Mathematik, 1

Tags: geometry , reflection , Locus

The figure shows a triangular pool table with sides $a$, $b$ and $c$. Located at point $S$ on $c$ a sphere - which can be assumed as a point. After kick-off, as indicated in the figure, it runs through as a result of reflections to $a, b, a, b$ and $c$ (in $S$) always the same track. The reflection occurs according to law of reflection. Characterize entilrely all triangles $ABC$, which allow such an orbit, and determine the locus of $S$. [img]https://cdn.artofproblemsolving.com/attachments/5/b/7662943e5b9ad321226e0c5f5daa3c4ac9faaa.png[/img]

1975 Chisinau City MO, 94

Tags: geometry , Locus

A straight line $\ell$ and a point $A$ outside of it are given on the plane. Find the locus of the vertices $C$ of the equilateral triangle $ABC$, the vertex $B$ of which lies on the straight line $\ell$.

1980 Spain Mathematical Olympiad, 7

Tags: geometry , analytic geometry , areas , Locus

The point $M$ varies on the segment $AB$ that measures $2$ m. a) Find the equation and the graphical representation of the locus of the points of the plane whose coordinates, $x$, and $y$, are, respectively, the areas of the squares of sides $AM$ and $MB$ . b) Find out what kind of curve it is. (Suggestion: make a $45^o$ axis rotation). c) Find the area of the enclosure between the curve obtained and the coordinate axes.

1948 Putnam, B2

Tags: Putnam , analytic geometry , Locus

A circle moves so that it is continually in the contact with all three coordinate planes of an ordinary rectangular system. Find the locus of the center of the circle.

1997 Czech And Slovak Olympiad IIIA, 6

Tags: Cyclic , parallelogram , diagonals , Locus , geometry

In a parallelogram $ABCD$, triangle $ABD$ is acute-angled and $\angle BAD = \pi /4$. Consider all possible choices of points $K,L,M,N$ on sides $AB,BC, CD,DA$ respectively, such that $KLMN$ is a cyclic quadrilateral whose circumradius equals those of triangles $ANK$ and $CLM$. Find the locus of the intersection of the diagonals of $KLMN$

2005 Sharygin Geometry Olympiad, 5

Tags: geometry , Locus , orthocenter , Circumcenter , Centroid , circumcircle

There are two parallel lines $p_1$ and $p_2$. Points $A$ and $B$ lie on $p_1$, and $C$ on $p_2$. We will move the segment $BC$ parallel to itself and consider all the triangles $AB'C '$ thus obtained. Find the locus of the points in these triangles: a) points of intersection of heights, b) the intersection points of the medians, c) the centers of the circumscribed circles.

1967 IMO Longlists, 9

Tags: geometry , incenter , angle bisector , Locus , Locus problems , IMO Shortlist , IMO Longlist

Circle $k$ and its diameter $AB$ are given. Find the locus of the centers of circles inscribed in the triangles having one vertex on $AB$ and two other vertices on $k.$

1978 Romania Team Selection Test, 5

Tags: geometry , Pure geometry , Locus

Find locus of points $ M $ inside an equilateral triangle $ ABC $ such that $$ \angle MBC+\angle MCA +\angle MAB={\pi}/{2}. $$

2022 Argentina National Olympiad, 3

Tags: geometry , Locus , Equilateral , midpoint

Given a square $ABCD$, let us consider an equilateral triangle $KLM$, whose vertices $K$, $L$ and $M$ belong to the sides $AB$, $BC$ and $CD$ respectively. Find the locus of the midpoints of the sides $KL$ for all possible equilateral triangles $KLM$. Note: The set of points that satisfy a property is called a locus.

1980 All Soviet Union Mathematical Olympiad, 287

Tags: geometry , areas , Locus , midpoints , convex

The points $M$ and $P$ are the midpoints of $[BC]$ and $[CD]$ sides of a convex quadrangle $ABCD$. It is known that $|AM| + |AP| = a$. Prove that $ABCD$ has area less than $\frac{a^2}{2}$.

Kyiv City MO 1984-93 - geometry, 1991.7.4

Tags: geometry , Locus , Equilateral

Given a circle, point $C$ on it and point $A$ outside the circle. The equilateral triangle $ACP$ is constructed on the segment $AC$. Point $C$ moves along the circle. What trajectory will the point $P$ describe?

2009 Switzerland - Final Round, 1

Tags: geometry , Locus , geometric inequality , hexagon

Let $P$ be a regular hexagon. For a point $A$, let $d_1\le d_2\le ...\le d_6$ the distances from $A$ to the six vertices of $P$, ordered by magnitude. Find the locus of all points $A$ in the interior or on the boundary of $P$ such that: (a) $d_3$ takes the smallest possible value. (b) $d_4$ takes the smallest possible value.

1986 Traian Lălescu, 2.2

Tags: analytic geometry , geometry , Locus , Locus problems

Let be a line $ d: 3x+4y-5=0 $ on a Cartesian plane. We mark with $ \mathcal{L} $ de locus of the planar points $ P $ such that the distance from $ P $ to $ d $ is double the distance from $ P $ to the origin. Let be $ B_{\lambda } ,C_{\lambda }\in\mathcal{L} $ such that $ C_{\lambda } -B_{\lambda } +\lambda =0. $ Find the locus of the middlepoints of the segments $ B_{\lambda }C_{\lambda }, $ if $ \lambda\in\mathbb{R} $ is variable.

2005 ISI B.Math Entrance Exam, 7

Tags: geometry , Locus , area of a triangle

Let $M$ be a point in the triangle $ABC$ such that \[\text{area}(ABM)=2 \cdot \text{area}(ACM)\] Show that the locus of all such points is a straight line.

V Soros Olympiad 1998 - 99 (Russia), 9.5

Tags: geometry , Locus , isosceles

An angle with vertex $A$ is given on the plane. Points $K$ and $P$ are selected on its sides so that $AK + AP = a$, where $a$ is a given segment. Let $M$ be a point on the plane such that the triangle $KPM$ is isosceles with the base $KP$ and the angle at the vertex $M$ equal to the given angle. Find the locus of points $M$ (as $K$ and $P$ move).

Ukrainian TYM Qualifying - geometry, XI.13

Tags: geometry , Locus , 3D geometry , cylinder , Ukrainian TYM

On the plane there are two cylindrical towers with radii of bases $r$ and $R$. Find the set of all those points of the plane from which these towers are visible at the same angle. Consider the case of more towers.

1966 IMO Shortlist, 16

Tags: geometry , Locus problems , Locus , circle , IMO Shortlist , IMO Longlist

We are given a circle $K$ with center $S$ and radius $1$ and a square $Q$ with center $M$ and side $2$. Let $XY$ be the hypotenuse of an isosceles right triangle $XY Z$. Describe the locus of points $Z$ as $X$ varies along $K$ and $Y$ varies along the boundary of $Q.$

2003 Federal Math Competition of S&M, Problem 4

Tags: geometry , Locus

An acute angle with the vertex $O$ and the rays $Op_1$ and $Op_2$ is given in a plane. Let $k_1$ be a circle with the center on $Op_1$ which is tangent to $Op_2$. Let $k_2$ be the circle that is tangent to both rays $Op_1$ and $Op_2$ and to the circle $k_1$ from outside. Find the locus of tangency points of $k_1$ and $k_2$ when center of $k_1$ moves along the ray $Op_1$.