Found problems: 68
2016 Chile National Olympiad, 6
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.
1969 IMO Longlists, 1
$(BEL 1)$ 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
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
$(HUN 6)$ 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.
1967 IMO Longlists, 9
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 $k.$
1986 Traian Lălescu, 2.2
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.
1966 IMO Shortlist, 16
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 $Q.$
2006 Sharygin Geometry Olympiad, 14
Given a circle and a fixed point $P$ not lying on it. Find the geometrical locus of the orthocenters of the triangles $ABP$, where $AB$ is the diameter of the circle.
1965 IMO, 5
Consider $\triangle OAB$ with acute angle $AOB$. Thorugh a point $M \neq O$ perpendiculars are drawn to $OA$ and $OB$, the feet of which are $P$ and $Q$ respectively. The point of intersection of the altitudes of $\triangle OPQ$ is $H$. What is the locus of $H$ if $M$ is permitted to range over
a) the side $AB$;
b) the interior of $\triangle OAB$.
2011 IMAR Test, 1
Let $A_0A_1A_2$ be a triangle and let $P$ be a point in the plane, not situated on the circle $A_0A_1A_2$. The line $PA_k$ meets again the circle $A_0A_1A_2$ at point $B_k, k = 0, 1, 2$. A line $\ell$ through the point $P$ meets the line $A_{k+1}A_{k+2}$ at point $C_k, k = 0, 1, 2$. Show that the lines $B_kC_k, k = 0, 1, 2$, are concurrent and determine the locus of their concurrency point as the line $\ell$ turns about the point $P$.
1973 IMO Shortlist, 1
Let a tetrahedron $ABCD$ be inscribed in a sphere $S$. Find the locus of points $P$ inside the sphere $S$ for which the equality
\[\frac{AP}{PA_1}+\frac{BP}{PB_1}+\frac{CP}{PC_1}+\frac{DP}{PD_1}=4\]
holds, where $A_1,B_1, C_1$, and $D_1$ are the intersection points of $S$ with the lines $AP,BP,CP$, and $DP$, respectively.
1965 IMO Shortlist, 5
Consider $\triangle OAB$ with acute angle $AOB$. Thorugh a point $M \neq O$ perpendiculars are drawn to $OA$ and $OB$, the feet of which are $P$ and $Q$ respectively. The point of intersection of the altitudes of $\triangle OPQ$ is $H$. What is the locus of $H$ if $M$ is permitted to range over
a) the side $AB$;
b) the interior of $\triangle OAB$.
2019 Oral Moscow Geometry Olympiad, 5
Given the segment $ PQ$ and a circle . A chord $AB$ moves around the circle, equal to $PQ$. Let $T$ be the intersection point of the perpendicular bisectors of the segments $AP$ and $BQ$. Prove that all points of $T$ thus obtained lie on one line.
1969 IMO Longlists, 12
$(CZS 1)$ Given a unit cube, find the locus of the centroids of all tetrahedra whose vertices lie on the sides of the cube.
1960 IMO, 5
Consider the cube $ABCDA'B'C'D'$ (with face $ABCD$ directly above face $A'B'C'D'$).
a) Find the locus of the midpoints of the segments $XY$, where $X$ is any point of $AC$ and $Y$ is any piont of $B'D'$;
b) Find the locus of points $Z$ which lie on the segment $XY$ of part a) with $ZY=2XZ$.
1963 IMO Shortlist, 2
Point $A$ and segment $BC$ are given. Determine the locus of points in space which are vertices of right angles with one side passing through $A$, and the other side intersecting segment $BC$.
1969 IMO Longlists, 39
$(HUN 6)$ 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.
1967 IMO Shortlist, 3
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 $k.$
1966 IMO Shortlist, 28
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.
2007 Sharygin Geometry Olympiad, 18
Determine the locus of vertices of triangles which have prescribed orthocenter and center of circumcircle.
1961 IMO, 6
Consider a plane $\epsilon$ and three non-collinear points $A,B,C$ on the same side of $\epsilon$; suppose the plane determined by these three points is not parallel to $\epsilon$. In plane $\epsilon$ take three arbitrary points $A',B',C'$. Let $L,M,N$ be the midpoints of segments $AA', BB', CC'$; Let $G$ be the centroid of the triangle $LMN$. (We will not consider positions of the points $A', B', C'$ such that the points $L,M,N$ do not form a triangle.) What is the locus of point $G$ as $A', B', C'$ range independently over the plane $\epsilon$?
2000 District Olympiad (Hunedoara), 3
Let $ \alpha $ be a plane and let $ ABC $ be an equilateral triangle situated on a parallel plane whose distance from $ \alpha $ is $ h. $ Find the locus of the points $ M\in\alpha $ for which
$$ \left|MA\right| ^2 +h^2 = \left|MB\right|^2 +\left|MC\right|^2. $$
1960 IMO, 7
An isosceles trapezoid with bases $a$ and $c$ and altitude $h$ is given.
a) On the axis of symmetry of this trapezoid, find all points $P$ such that both legs of the trapezoid subtend right angles at $P$;
b) Calculate the distance of $p$ from either base;
c) Determine under what conditions such points $P$ actually exist. Discuss various cases that might arise.
1966 IMO Longlists, 17
Let $ABCD$ and $A^{\prime }B^{\prime}C^{\prime }D^{\prime }$ be two arbitrary parallelograms in the space, and let $M,$ $N,$ $P,$ $Q$ be points dividing the segments $AA^{\prime },$ $BB^{\prime },$ $CC^{\prime },$ $DD^{\prime }$ in equal ratios.
[b]a.)[/b] Prove that the quadrilateral $MNPQ$ is a parallelogram.
[b]b.)[/b] What is the locus of the center of the parallelogram $MNPQ,$ when the point $M$ moves on the segment $AA^{\prime }$ ?
(Consecutive vertices of the parallelograms are labelled in alphabetical order.
1986 IMO Shortlist, 1
Let $A,B$ be adjacent vertices of a regular $n$-gon ($n\ge5$) with center $O$. A triangle $XYZ$, which is congruent to and initially coincides with $OAB$, moves in the plane in such a way that $Y$ and $Z$ each trace out the whole boundary of the polygon, with $X$ remaining inside the polygon. Find the locus of $X$.