Found problems: 288
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 $\triangle OAB$.
2017 Sharygin Geometry Olympiad, P5
A segment $AB$ is fixed on the plane. Consider all acute-angled triangles with side $AB$. Find the locus of
а) the vertices of their greatest angles,
b) their incenters.
2006 Sharygin Geometry Olympiad, 24
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
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 $P$.
1953 Putnam, B6
Let $P$ and $Q$ be any points inside a circle $C$ with center $O$ such that $OP=OQ.$ Determine the location of a point $Z$ on $C$ such that $PZ+QZ$ is minimal.
1974 Czech and Slovak Olympiad III A, 6
Let a unit square $\mathcal D$ be given in the plane. For any point $X$ in the plane denote $\mathcal D_X$ the image of $\mathcal D$ in rotation with respect to origin $X$ by $+90^\circ.$ Find the locus of all $X$ such that the area of union $\mathcal D\cup\mathcal D_X$ is at most 1.5.
1965 Vietnam National Olympiad, 2
$AB$ and $CD$ are two fixed parallel chords of the circle $S$. $M$ is a variable point on the circle. $Q$ is the intersection of the lines $MD$ and $AB$. $X$ is the circumcenter of the triangle $MCQ$.
Find the locus of $X$.
What happens to $X$ as $M$ tends to
(1) $D$,
(2) $C$?
Find a point $E$ outside the plane of $S$ such that the circumcenter of the tetrahedron $MCQE$ has the same locus as $X$.
1986 Bulgaria National Olympiad, Problem 5
Let $A$ be a fixed point on a circle $k$. Let $B$ be any point on $k$ and $M$ be a point such that $AM:AB=m$ and $\angle BAM=\alpha$, where $m$ and $\alpha$ are given. Find the locus of point $M$ when $B$ describes the circle $k$.
1982 Austrian-Polish Competition, 2
Let $F$ be a closed convex region inside a circle $C$ with center $O$ and radius $1$. Furthermore, assume that from each point of $C$ one can draw two rays tangent to $F$ which form an angle of $60^o$. Prove that $F$ is the disc centered at $O$ with radius $1/2$.
2011 Sharygin Geometry Olympiad, 13
a) Find the locus of centroids for triangles whose vertices lie on the sides of a given triangle (each side contains a single vertex).
b) Find the locus of centroids for tetrahedrons whose vertices lie on the faces of a given tetrahedron (each face contains a single vertex).
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.
1977 Czech and Slovak Olympiad III A, 3
Consider any complex units $Z,W$ with $\text{Im}\ Z\ge0,\text{Re}\,W\ge 0.$ Determine and draw the locus of all possible sums $S=Z+W$ in the complex plane.
1973 IMO Shortlist, 2
Given a circle $K$, find the locus of vertices $A$ of parallelograms $ABCD$ with diagonals $AC \leq BD$, such that $BD$ is inside $K$.
2007 Sharygin Geometry Olympiad, 4
Given a triangle $ABC$. An arbitrary point $P$ is chosen on the circumcircle of triangle $ABH$ ($H$ is the orthocenter of triangle $ABC$). Lines $AP$ and $BP$ meet the opposite sidelines of the triangle at points $A' $ and $B'$, respectively. Determine the locus of midpoints of segments $A'B'$.
2006 Sharygin Geometry Olympiad, 10.3
Given a circle and a point $P$ inside it, different from the center. We consider pairs of circles tangent to the given internally and to each other at point $P$. Find the locus of the points of intersection of the common external tangents to these circles.
2004 Oral Moscow Geometry Olympiad, 3
Given a square $ABCD$. Find the locus of points $M$ such that $\angle AMB = \angle CMD$.
1980 Poland - Second Round, 3
There is a sphere $ K $ in space and points $ A, B $ outside the sphere such that the segment $ AB $ intersects the interior of the sphere. Prove that the set of points $ P $ for which the segments $ AP $ and $ BP $ are tangent to the sphere $ K $ is contained in a certain plane.
1992 IMO Longlists, 17
In the plane let $\,C\,$ be a circle, $\,L\,$ a line tangent to the circle $\,C,\,$ and $\,M\,$ a point on $\,L$. Find the locus of all points $\,P\,$ with the following property: there exists two points $\,Q,R\,$ on $\,L\,$ such that $\,M\,$ is the midpoint of $\,QR\,$ and $\,C\,$ is the inscribed circle of triangle $\,PQR$.
1958 Czech and Slovak Olympiad III A, 4
Consider positive numbers $d,v$ such that $d>v$. Moreover, consider two perpendicular skew lines $p,q$ of distance $v$ (that is direction vectors of both lines are orthogonal and $\min_{X\in p,Y\in q}XY = v$). Finally, consider all line segments $PQ$ such that $P\in p, Q\in q, PQ=d$.
a) Find the locus of all points $P$.
b) Find the locus of all midpoints of segments $PQ$.
2020 Tournament Of Towns, 5
Given are two circles which intersect at points $P$ and $Q$. Consider an arbitrary line $\ell$ through $Q$, let the second points of intersection of this line with the circles be $A$ and $B$ respectively. Let $C$ be the point of intersection of the tangents to the circles in those points. Let $D$ be the intersection of the line $AB$ and the bisector of the angle $CPQ$. Prove that all possible $D$ for any choice of $\ell$ lie on a single circle.
Alexey Zaslavsky
2021 Stars of Mathematics, 3
Let $ABC$ be a triangle, let its $A$-symmedian cross the circle $ABC$ again at $D$, and let $Q$ and $R$ be the feet of the perpendiculars from $D$ on the lines $AC$ and $AB$, respectively. Consider a variable point $X$ on the line $QR$, different from both $Q$ and $R$. The line through $X$ and perpendicular to $DX$ crosses the lines $AC$ and $AB$ at $V$ and $W$, respectively. Determine the geometric locus of the midpoint of the segment $VW$.
[i]Adapted from American Mathematical Monthly[/i]
1977 Bulgaria National Olympiad, Problem 4
Vertices $A$ and $C$ of the quadrilateral $ABCD$ are fixed points of the circle $k$ and each of the vertices $B$ and $D$ is moving to one of the arcs of $k$ with ends $A$ and $C$ in such a way that $BC=CD$. Let $M$ be the intersection point of $AC$ and $BD$ and $F$ is the center of the circumscribed circle around $\triangle ABM$. Prove that the locus of $F$ is an arc of a circle.
[i]J. Tabov[/i]
2000 Austrian-Polish Competition, 7
Triangle $A_0B_0C_0$ is given in the plane. Consider all triangles $ABC$ such that:
(i) The lines $AB,BC,CA$ pass through $C_0,A_0,B_0$, respectvely,
(ii) The triangles $ABC$ and $A_0B_0C_0$ are similar.
Find the possible positions of the circumcenter of triangle $ABC$.
1989 Romania Team Selection Test, 3
Let $F$ be the boundary and $M,N$ be any interior points of a triangle $ABC$. Consider the function $f_{M,N}: F \to R$ defined by $f_{M,N}(P) = MP^2 +NP^2$ and let $\eta_{M,N}$ be the number of points $P$ for which $f{M,N}$ attains its minimum.
(a) Prove that $1 \le \eta_{M,N} \le 3$.
(b) If $M$ is fixed, find the locus of $N$ for which $\eta_{M,N} > 1$.
(c) Prove that the locus of $M$ for which there are points $N$ such that $\eta_{M,N} = 3$ is the interior of a tangent hexagon.
2005 Sharygin Geometry Olympiad, 5
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.