Found problems: 288
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 $XB$.
1960 IMO Shortlist, 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.
1969 Spain Mathematical Olympiad, 2
Find the locus of the affix $M$, of the complex number $z$, so that it is aligned with the affixes of $i$ and $iz$ .
2010 Cuba MO, 8
Let $ABCDE$ be a convex pentagon that has $AB < BC$, $AE <ED$ and $AB + CD + EA = BC + DE$. Variable points $F,G$ and $H$ are taken that move on the segments $BC$, $CD$ and $OF$ respectively . $B'$ is defined as the projection of $B$ on $AF$, $C'$ as the projection of $C$ on $FG$, $D'$ as the projection of $D$ on $GH$ and $E'$ as the projection of $E$ onto $HA$. Prove that there is at least one quadrilateral $B'C'D'E'$ when $F,G$ and $H$ move on their sides, which is a parallelogram.
2008 Grigore Moisil Intercounty, 2
Given a convex quadrilateral $ ABCD, $ find the locus of points $ X $ that verify the qualities:
$$ XA^2+XB^2+CD^2=XB^2+XC^2+DA^2=XC^2+XD^2+AB^2=XD^2+XA^2+BC^2 $$
[i]Maria Pop[/i]
1962 All-Soviet Union Olympiad, 2
Given a fixed circle $C$ and a line L through the center $O$ of $C$. Take a variable point $P$ on $L$ and let $K$ be the circle with center $P$ through $O$. Let $T$ be the point where a common tangent to $C$ and $K$ meets $K$. What is the locus of $T$?
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
1955 Moscow Mathematical Olympiad, 297
Given two distinct nonintersecting circles none of which is inside the other.
Find the locus of the midpoints of all segments whose endpoints lie on the circles.
Estonia Open Senior - geometry, 1995.2.4
Find all points on the plane such that the sum of the distances of each of the four lines defined by the unit square of that plane is $4$.
2014 Bulgaria National Olympiad, 1
Let $k$ be a given circle and $A$ is a fixed point outside $k$. $BC$ is a diameter of $k$. Find the locus of the orthocentre of $\triangle ABC$ when $BC$ varies.
[i]Proposed by T. Vitanov, E. Kolev[/i]
1942 Putnam, A1
A square of side $2a$, lying always in the first quadrant of the $xy$-plane, moves so that two consecutive vertices
are always on the $x$- and $y$-axes respectively. Find the locus of the midpoint of the square.
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.
1995 Grosman Memorial Mathematical Olympiad, 4
Two given circles $\alpha$ and $\beta$ intersect each other at two points.
Find the locus of the centers of all circles that are orthogonal to both $\alpha$ and $\beta$.
2022 Oral Moscow Geometry Olympiad, 5
Given a circle and a straight line $AB$ passing through its center (points $A$ and $B$ are fixed, $A$ is outside the circle, and $B$ is inside). Find the locus of the intersection of lines $AX$ and $BY$, where $XY$ is an arbitrary diameter of the circle.
(A. Akopyan, A. Zaslavsky)
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.
2004 Tournament Of Towns, 4
A circle with the center $I$ is entirely inside of a circle with center $O$. Consider all possible chords $AB$ of the larger circle which are tangent to the smaller one. Find the locus of the centers of the circles circumscribed about the triangle $AIB$.
2014 Contests, 1
Let $k$ be a given circle and $A$ is a fixed point outside $k$. $BC$ is a diameter of $k$. Find the locus of the orthocentre of $\triangle ABC$ when $BC$ varies.
[i]Proposed by T. Vitanov, E. Kolev[/i]
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 $\,PQR$.
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.
1979 Bundeswettbewerb Mathematik, 2
A circle $k$ with center $M$ and radius $r$ is given. Find the locus of the incenters of all obtuse-angled triangles inscribed in $k$.
1959 Polish MO Finals, 5
In the plane of the triangle $ ABC $ a straight line moves which intersects the sides $ AC $ and $ BC $ in such points $ D $ and $ E $ that $ AD = BE $. Find the locus of the midpoint $ M $ of the segment $ DE $.
1982 Spain Mathematical Olympiad, 8
Given a set $C$ of points in the plane, it is called the distance of a point $P$ from the plane to the set $C$ at the smallest of the distances from $P$ to each of the points of $C$. Let the sets be $C = \{A,B\}$, with $A = (1, 0)$ and $B = (2, 0)$; and $C'= \{A',B'\}$ with $A' = (0, 1)$ and $B' = (0, 7)$, in an orthogonal reference system. Find and draw the set $M$ of points in the plane that are equidistant from $C$ and $C'$ . Study whether the function whose graph is the set $M$ previously obtained is derivable.
2007 Sharygin Geometry Olympiad, 4
Determine the locus of orthocenters of triangles, given the midpoint of a side and the feet of the altitudes drawn on two other sides.
1999 Estonia National Olympiad, 5
Let $C$ be an interior point of line segment $AB$. Equilateral triangles $ADC$ and $CEB$ are constructed to the same side from $AB$. Find all points which can be the midpoint of the segment $DE$.
1995 Tournament Of Towns, (477) 1
If P is a point inside a convex quadrilateral $ABCD$, let the angle bisectors of $\angle APB$, $\angle BPC$, $\angle CPD$ and $\angle DPA$ meet $AB$, $BC$, $CD$ and $DA$ at $K$, $L$, $M$ and $N$ respectively.
(a) Find a point $P$ such that $KLMN$ is a parallelogram.
(b) Find the locus of all such points $P$.
(S Tokarev)