This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 288

1995 Grosman Memorial Mathematical Olympiad, 4
Tags: circles , locus , geometry , intersecting circles
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 $\beta$.

Ukrainian TYM Qualifying - geometry, 2015.23
Tags: geometry , locus
An acute-angled triangle $ABC$ is given, through the vertices $B$ and $C$ of which a circle $\Omega$, $A \notin \Omega$, is drawn. We consider all points $P \in \Omega$, that do not lie on none of the lines $AB$ and $AC$ and for which the common tangents of the circumscribed circles of triangles $APB$ and $APC$ are not parallel. Let $X_P$ be the point of intersection of such two common tangents. a) Prove that the locus of points $X_P$ lies to some two lines. b) Prove that if the circle $\Omega$ passes through the orthocenter of the triangle $ABC$, then one of these lines is the line $BC$.

1949-56 Chisinau City MO, 32
Tags: midpoint , geometry , locus , right triangle
Determine the locus of points that are the midpoints of segments of equal length, the ends of which lie on the sides of a given right angle.

1963 IMO Shortlist, 2
Tags: geometry , 3d geometry , sphere , locus , locus problems
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 $BC$.

1987 Tournament Of Towns, (148) 5
Tags: geometry , locus , perpendicular , equilateral
Perpendiculars are drawn from an interior point $M$ of the equilateral triangle $ABC$ to its sides , intersecting them at points $D, E$ and $F$ . Find the locus of all points $M$ such that $DEF$ is a right triangle . (J . Tabov , Sofia)

1957 Moscow Mathematical Olympiad, 359
Tags: geometry , locus , perpendicular
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.

1969 IMO Longlists, 53
Tags: trigonometry , geometry , locus , locus problems
$(POL 2)$ Given two segments $AB$ and $CD$ not in the same plane, find the locus of points $M$ such that $MA^2 +MB^2 = MC^2 +MD^2.$

1986 IMO Longlists, 33
Tags: geometry , rotation , polygon , locus , locus problems
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 $X$.

1960 IMO Shortlist, 5
Tags: geometry , 3d geometry , cube , locus , locus problems
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 $ZY=2XZ$.

2018 Peru Cono Sur TST, 7
Tags: locus , geometry
Let $ABCD$ be a fixed square and $K$ a variable point on segment $AD$. The square $KLMN$ is constructed such that $B$ is on segment $LM$ and $C$ is on segment $MN$. Let $T$ be the intersection point of lines $LA$ and $ND$. Find the locus of $T$ as $K$ varies along segment $AD$.

2022 Belarusian National Olympiad, 11.4
Tags: geometry , locus
On plane circles $\omega_1, \omega_2, \omega_3$ with centers $O_1,O_2,O_3$ are given such that $\omega_1$ is externally tangent $\omega_2$ and $\omega_3$ at points $P, Q$ respectively. On $\omega_1$ point $C$ is chosen arbitrarily. Line $CP$ intersects $\omega_2$ at $B$, line $CQ$ intersects $\omega_3$ at $A$. Point $O$ is the circumcenter of $ABC$. Prove that the locus of points $O$ (when $C$ changes) is a circle, the center of which lies on the circumcircle of $O_1O_2O_3$

2016 Tournament Of Towns, 2
Tags: geometry , locus , circles
On plane there is fixed ray $s$ with vertex $A$ and a point $P$ not on the line which contains $s$. We choose a random point $K$ which lies on ray. Let $N$ be a point on a ray outside $AK$ such that $NK=1$. Let $M$ be a point such that $NM=1,M \in PK$ and $M!=K.$ Prove that all lines $NM$, provided by some point $K$, touch some fixed circle.

1969 Spain Mathematical Olympiad, 1
Tags: geometry , locus , inversion
Find the locus of the centers of the inversions that transform two points $A, B$ of a given circle $\gamma$ , at diametrically opposite points of the inverse circles of $\gamma$ .

2000 Regional Competition For Advanced Students, 3
Tags: locus , midpoint , geometry
We consider two circles $k_1(M_1, r_1)$ and $k_2(M_2, r_2)$ with $z = M_1M_2 > r_1+r_2$ and a common outer tangent with the tangent points $P_1$ and $P2$ (that is, they lie on the same side of the connecting line $M_1M_2$). We now change the radii so that their sum is $r_1+r_2 = c$ remains constant. What set of points does the midpoint of the tangent segment $P_1P_2$ run through, when $r_1$ varies from $0$ to $c$?

2014 Nordic, 2
Tags: geometry , distance , locus , equilateral triangle
Given an equilateral triangle, find all points inside the triangle such that the distance from the point to one of the sides is equal to the geometric mean of the distances from the point to the other two sides of the triangle.

1967 German National Olympiad, 1
Tags: geometry , locus , equilateral , square
In a plane, a square $ABCD$ and a point $P$ located inside it are given. Let a point $ Q$ pass through all sides of the square. Describe the set of all those points $R$ in for which the triangle $PQR$ is equilateral.

2007 Sharygin Geometry Olympiad, 18
Tags: geometry , locus , locus problems , orthocenter , circumcenter , circumcircle
Determine the locus of vertices of triangles which have prescribed orthocenter and center of circumcircle.

2013 Sharygin Geometry Olympiad, 7
Tags: geometry , locus , collinear , circles
Two fixed circles $\omega_1$ and $\omega_2$ pass through point $O$. A circle of an arbitrary radius $R$ centered at $O$ meets $\omega_1$ at points $A$ and $B$, and meets $\omega_2$ at points $C$ and $D$. Let $X$ be the common point of lines $AC$ and $BD$. Prove that all the points X are collinear as $R$ changes.

2018 Bundeswettbewerb Mathematik, 3
Tags: perpendicular lines , locus , geometry
Let $T$ be a point on a line segment $AB$ such that $T$ is closer to $B$ than to $A$. Show that for each point $C \ne T$ on the line through $T$ perpendicular to $AB$ there is exactly one point $D$ on the line segment $AC$ with $\angle CBD=\angle BAC$. Moreover, show that the line through $D$ perpendicular to $AC$ intersects the line $AB$ in a point $E$ which is independent of the position of $C$.

1974 Putnam, A5
Tags: locus , parabola , geometry
Consider the two mutually tangent parabolas $y=x^2$ and $y=-x^2$. The upper parabola rolls without slipping around the fixed lower parabola. Find the locus of the focus of the moving parabola.

Kyiv City MO Seniors 2003+ geometry, 2016.10.4
Tags: geometry , locus , collinear , circles
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)

1998 Belarus Team Selection Test, 1
Tags: geometry , locus , circles , angle
Two circles $S_1$ and $S_2$ intersect at different points $P,Q$. The arc of $S_1$ lying inside $S_2$ measures $2a$ and the arc of $S_2$ lying inside $S_1$ measures $2b$. Let $T$ be any point on $S_1$. Let $R,S$ be another points of intersection of $S_2$ with $TP$ and $TQ$ respectively. Let $a+2b<\pi$ . Find the locus of the intersection points of $PS$ and $RQ$. S.Shikh

2016 Puerto Rico Team Selection Test, 5
Tags: geometry , locus , area
$ABCD$ is a quadrilateral, $E, F, G, H$ are the midpoints of $AB$, $BC$, $CD$, $DA$ respectively. Find the point $P$ such that area $(PHAE) =$ area $(PEBF) =$ area $(PFCG) =$ area $(PGDH).$

1971 Czech and Slovak Olympiad III A, 5
Tags: triangle , geometry , equilateral , locus
Let $ABC$ be a given triangle. Find the locus $\mathbf M$ of all vertices $Z$ such that triangle $XYZ$ is equilateral where $X$ is any point of segment $AB$ and $Y\neq X$ lies on ray $AC.$

1924 Eotvos Mathematical Competition, 2
Tags: geometry , locus , fixed
If $O$ is a given point, $\ell$ a given line, and $a$ a given positive number, find the locus of points $P$ for which the sum of the distances from $P$ to $O$ and from $P$ to $\ell$ is $a$.