Found problems: 288
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]
1996 Abels Math Contest (Norwegian MO), 1
Let $S$ be a circle with center $C$ and radius $r$, and let $P \ne C$ be an arbitrary point.
A line $\ell$ through $P$ intersects the circle in $X$ and $Y$. Let $Z$ be the midpoint of $XY$.
Prove that the points $Z$, as $\ell$ varies, describe a circle. Find the center and radius of this circle.
1948 Putnam, B2
A circle moves so that it is continually in the contact with all three coordinate planes of an ordinary rectangular system. Find the locus of the center of the circle.
Kyiv City MO Seniors 2003+ geometry, 2016.10.4
On the circle with diameter $AB$, the point $M$ was selected and fixed. Then the point ${{Q} _ {i}}$ is selected, for which the chord $M {{Q} _ {i}}$ intersects $AB$ at the point ${{K} _ {i}}$ and thus $ \angle M {{K} _ {i}} B <90 {} ^ \circ$. A chord that is perpendicular to $AB$ and passes through the point ${{K} _ {i}}$ intersects the line $B {{Q} _ {i}}$ at the point ${{P } _ {i}}$. Prove that the points ${{P} _ {i}}$ in all possible choices of the point ${{Q} _ {i}}$ lie on the same line.
(Igor Nagel)
Ukrainian TYM Qualifying - geometry, 2013.6
Given a circle $\omega$, on which marks the points $A,B,C$. Let $BF$ and $CE$ be the altitudes of the triangle $ABC$, $M$ be the midpoint of the side $AC$. Find a the locus of the intersection points of the lines $BF$ and E$M$ for all positions of point $A$ , as $A$ moves on $\omega$.
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 $BC$.
1962 Poland - Second Round, 5
In the plane there is a square $ Q $ and a point $ P $. The point $ K $ runs through the perimeter of the square $ Q $. Find the locus of the vertex $ M $ of the equilateral triangle $ KPM $.
1960 IMO Shortlist, 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 $ZY=2XZ$.
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 $\epsilon$?
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 $P$.
Ukrainian From Tasks to Tasks - geometry, 2012.2
The triangle $ABC$ is equilateral. Find the locus of the points $M$ such that the triangles $ABM$ and $ACM$ are both isosceles.
1986 IMO, 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 $X$.
1981 Czech and Slovak Olympiad III A, 3
Let $ABCD$ be a unit square. Consider an equilateral triangle $XYZ$ with $X,Y$ as (inner or boundary) points of the square. Determine the locus $M$ of vertices $Z$ of all these triangles $XYZ$ and compute the area of $M.$
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.
1949 Moscow Mathematical Olympiad, 165
Consider two triangles, $ABC$ and $DEF$, and any point $O$. We take any point $X$ in $\vartriangle ABC$ and any point $Y$ in $\vartriangle DEF$ and draw a parallelogram $OXY Z$. Prove that the locus of all possible points $Z$ form a polygon. How many sides can it have? Prove that its perimeter is equal to the sum of perimeters of the original triangles.
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.
1994 Spain Mathematical Olympiad, 2
Let $Oxyz$ be a trihedron whose edges $x,y, z$ are mutually perpendicular. Let $C$ be the point on the ray $z$ with $OC = c$. Points $P$ and $Q$ vary on the rays $x$ and $y$ respectively in such a way that $OP+OQ = k$ is constant. For every $P$ and $Q$, the circumcenter of the sphere through $O,C,P,Q$ is denoted by $W$. Find the locus of the projection of $W$ on the plane O$xy$. Also find the locus of points $W$.
1969 IMO Longlists, 4
$(BEL 4)$ Let $O$ be a point on a nondegenerate conic. A right angle with vertex $O$ intersects the conic at points $A$ and $B$. Prove that the line $AB$ passes through a fixed point located on the normal to the conic through the point $O.$
2021/2022 Tournament of Towns, P4
Given is a segment $AB$. Three points $X, Y, Z$ are picked in the space so that $ABX$ is an equilateral triangle and $ABYZ$ is a square. Prove that the orthocenters of all triangles $XYZ$ obtained in this way belong to a fixed circle.
[i]Alexandr Matveev[/i]
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.
2001 Taiwan National Olympiad, 4
Let $\Gamma$ be the circumcircle of a fixed triangle $ABC$, and let $M$ and $N$ be the midpoints of the arcs $BC$ and $CA$, respectively. For any point $X$ on the arc $AB$, let $O_1$ and $O_2$ be the incenters of $\vartriangle XAC$ and $\vartriangle XBC$, and let the circumcircle of $\vartriangle XO_1O_2$ intersect $\Gamma$ at $X$ and $Q$. Prove that triangles $QNO_1$ and $QMO_2$ are similar, and find all possible locations of point $Q$.
2019 Argentina National Olympiad Level 2, 3
Let $\Gamma$ be a circle of center $S$ and radius $r$ and let be $A$ a point outside the circle. Let $BC$ be a diameter of $\Gamma$ such that $B$ does not belong to the line $AS$ and consider the point $O$ where the perpendicular bisectors of triangle $ABC$ intersect, that is, the circumcenter of $ABC$. Determine all possible locations of point $O$ when $B$ varies in circle $\Gamma$.
1957 Moscow Mathematical Olympiad, 359
Straight lines $OA$ and $OB$ are perpendicular. Find the locus of endpoints $M$ of all broken lines $OM$ of length $\ell$ which intersect each line parallel to $OA$ or $OB$ at not more than one point.
VII Soros Olympiad 2000 - 01, 9.8
Given a triangle $ABC$. On its sides $BC$ , $CA$ and $AB$ , the points $A_1$ , $B_1$ and $C_1$ are taken, respectively , such that
$2 \angle B_1 A_1 C_1 + \angle BAC = 180^o$ ,
$2 \angle A_1 C_1 B_1 + \angle ACB = 180^o$ ,
$2 \angle C_1 B_1 A_1 + \angle CBA = 180^o$ .
Find the locus of the centers of the circles circumscribed about the triangles $A_1 B_1 C_1$ (all possible such triangles are considered).
2015 Sharygin Geometry Olympiad, P20
Given are a circle and an ellipse lying inside it with focus $C$. Find the locus of the circumcenters of triangles $ABC$, where $AB$ is a chord of the circle touching the ellipse.