Found problems: 288
1987 Mexico National Olympiad, 3
Consider two lines $\ell$ and $\ell ' $ and a fixed point $P$ equidistant from these lines. What is the locus of projections $M$ of $P$ on $AB$, where $A$ is on $\ell $, $B$ on $\ell ' $, and angle $\angle APB$ is right?
1940 Moscow Mathematical Olympiad, 061
Given two lines on a plane, find the locus of all points with the difference between the distance to one line and the distance to the other equal to the length of a given segment.
VII Soros Olympiad 2000 - 01, 10.5
An acute-angled triangle $ABC$ is given. Points $A_1, B_1$ and $C_1$ are taken on its sides $BC, CA$ and $AB$, respectively, such that
$\angle B_1A_1C_1 + 2 \angle BAC = 180^o$,
$\angle A_1C_1B_1 + 2 \angle ACB = 180^o$,
$\angle C_1B_1A_1 + 2 \angle CBA = 180^o$.
Find the locus of the centers of the circles inscribed in triangles $A_1B_1C_1$ (all kinds of such triangles are considered).
1978 IMO, 2
We consider a fixed point $P$ in the interior of a fixed sphere$.$ We construct three segments $PA, PB,PC$, perpendicular two by two$,$ with the vertexes $A, B, C$ on the sphere$.$ We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as edges$.$ Find the locus of the point $Q$ when $A, B, C$ take all the positions compatible with our problem.
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.
1952 Moscow Mathematical Olympiad, 224-
You are given a segment $AB$. Find the locus of the vertices $C$ of acute-angled triangles $ABC$.
1975 Czech and Slovak Olympiad III A, 5
Let a square $\mathbf P=P_1P_2P_3P_4$ be given in the plane. Determine the locus of all vertices $A$ of isosceles triangles $ABC,AB=BC$ such that the vertices $B,C$ are points of the square $\mathbf P.$
1973 IMO Shortlist, 5
A circle of radius 1 is located in a right-angled trihedron and touches all its faces. Find the locus of centers of such circles.
1953 Moscow Mathematical Olympiad, 239
On the plane find the locus of points whose coordinates satisfy $sin(x + y) = 0$.
2002 Junior Balkan Team Selection Tests - Moldova, 7
The side of the square $ABCD$ has a length equal to $1$. On the sides $(BC)$ ¸and $(CD)$ take respectively the arbitrary points $M$ and $N$ so that the perimeter of the triangle $MCN$ is equal to $2$.
a) Determine the measure of the angle $\angle MAN$.
b) If the point $P$ is the foot of the perpendicular taken from point $A$ to the line $MN$, determine the locus of the points $P$.
1999 Greece JBMO TST, 5
$\Phi$ is the union of all triangles that are symmetric of the triangle $ABC$ wrt a point $O$, as point $O$ moves along the triangle's sides. If the area of the triangle is $E$, find the area of $\Phi$.
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.
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 $\epsilon$?