Found problems: 288
1945 Moscow Mathematical Olympiad, 098
A right triangle $ABC$ moves along the plane so that the vertices $B$ and $C$ of the triangle’s acute angles slide along the sides of a given right angle. Prove that point $A$ fills in a line segment and find its length.
Champions Tournament Seniors - geometry, 2007.3
Given a triangle $ABC$. Point $M$ moves along the side $BA$ and point $N$ moves along the side $AC$ beyond point $C$ such that $BM=CN$. Find the geometric locus of the centers of the circles circumscribed around the triangle $AMN$.
V Soros Olympiad 1998 - 99 (Russia), 9.5
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).
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 $k.$
2007 Hanoi Open Mathematics Competitions, 8
Let $ABC$ be an equilateral triangle. For a point $M$ inside $\vartriangle ABC$, let $D,E,F$ be the feet of the perpendiculars from $M$ onto $BC,CA,AB$, respectively. Find the locus of all such points $M$ for which $\angle FDE$ is a right angle.
III Soros Olympiad 1996 - 97 (Russia), 10.7
Let $A$ be a fixed point on a circle, $B$ and$ C$ be arbitrary points on the circle different from $A$ and at different distances. The bisector of the angle $\angle BAC$ intersects the chord $BC$ and the circle at points $K$ and $P$, $D$ is the projection of $A$ onto the straight line $BC$. A circle passing through points $K$, $P$ and $D$ intersects the straight line $AD$ for the second time at point $M$. Find the locus of points $M$.
1995 Bundeswettbewerb Mathematik, 2
A line $g$ and a point $A$ outside $g$ are given in a plane. A point $P$ moves along $g$.
Find the locus of the third vertices of equilateral triangles whose two vertices are $A$ and $P$.
1989 Swedish Mathematical Competition, 4
Let $ABCD$ be a regular tetrahedron. Find the positions of point $P$ on the edge $BD$ such that the edge $CD$ is tangent to the sphere with diameter $AP$.
Estonia Open Junior - geometry, 1999.1.2
Two different points $X$ and $Y$ are chosen in the plane. Find all the points $Z$ in this plane for which the triangle $XYZ$ is isosceles.
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.
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 $\triangle OAB$.
2017 Yasinsky Geometry Olympiad, 6
Given a circle $\omega$ of radius $r$ and a point $A$, which is far from the center of the circle at a distance $d<r$. Find the geometric locus of vertices $C$ of all possible $ABCD$ rectangles, where points $B$ and $D$ lie on the circle $\omega$.
2009 Switzerland - Final Round, 1
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.
1941 Moscow Mathematical Olympiad, 074
A point $P$ lies outside a circle. Consider all possible lines drawn through $P$ so that they intersect the circle. Find the locus of the midpoints of the chords — segments the circle intercepts on these lines.
2012 Sharygin Geometry Olympiad, 4
Consider a square. Find the locus of midpoints of the hypothenuses of rightangled triangles with the vertices lying on three different sides of the square and not coinciding with its vertices.
(B.Frenkin)
1992 IMO Shortlist, 20
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 $\,PQR$.
2022 Argentina National Olympiad, 3
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.
1963 Dutch Mathematical Olympiad, 2
The straight lines $k$ and $\ell$ intersect at right angles. A line intersects $k$ in $A$ and $\ell$ in $B$. Consider all straight line segments $PQ$ ($P$ on $k$ and $Q$ on $\ell$), which makes an angle of $45^o$ with $AB$.
(a) Determine the locus of the midpoints of the line segments $PQ$,
(b) If the perpendicular bisector of such a line segment $PQ$ intersects the line $k$ at $K$ and the line $\ell$ at $L$, then prove that $KL \ge PQ$.
[hide=original wording of second sentence]De loodrechte snijlijn van k en l snijdt k in A en t in B[/hide]
Ukrainian TYM Qualifying - geometry, 2011.5
The circle $\omega_0$ touches the line at point A. Let $R$ be a given positive number. We consider various circles $\omega$ of radius $R$ that touch a line $\ell$ and have two different points in common with the circle $\omega_0$. Let $D$ be the touchpoint of the circle $\omega_0$ with the line $\ell$, and the points of intersection of the circles $\omega$ and $\omega_0$ are denoted by $B$ and $C$ (Assume that the distance from point $B$ to the line $\ell$ is greater than the distance from point $C$ to this line). Find the locus of the centers of the circumscribed circles of all such triangles $ABD$.
1956 Moscow Mathematical Olympiad, 325
On sides $AB$ and $CB$ of $\vartriangle ABC$ there are drawn equal segments, $AD$ and $CE$, respectively, of arbitrary length (but shorter than min($AB,BC$)). Find the locus of midpoints of all possible segments $DE$.
Kvant 2024, M2784
The bisectors $AD{}$ and $BE{}$ were drawn in the triangle $ABC{}$ and they intersected at point $I{}.$ Then everything was erased, leaving only the points $D{}$ and $E{}.$ Find the set of possible positions of the point $I{}.$
[i]Proposed by M. Didin[/i]
2006 Sharygin Geometry Olympiad, 13
Two straight lines $a$ and $b$ are given and also points $A$ and $B$. Point $X$ slides along the line $a$, and point $Y$ slides along the line $b$, so that $AX \parallel BY$. Find the locus of the intersection point of $AY$ with $XB$.
1953 Moscow Mathematical Olympiad, 239
On the plane find the locus of points whose coordinates satisfy $sin(x + y) = 0$.
1957 Kurschak Competition, 1
$ABC$ is an acute-angled triangle. $D$ is a variable point in space such that all faces of the tetrahedron $ABCD$ are acute-angled. $P$ is the foot of the perpendicular from $D$ to the plane $ABC$. Find the locus of $P$ as $D$ varies.
2002 Moldova Team Selection Test, 3
A triangle $ABC$ is inscribed in a circle $G$. For any point $M$ inside the triangle, $A_1$ denotes the intersection of the ray $AM$ with $G$. Find the locus of point $M$ for which $\frac{BM\cdot CM}{MA_1}$ is minimal, and find this minimum value.