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.

AND:
OR:
NO:

Found problems: 288

1960 Putnam, A4
Tags: locus , geometry
Given two points, $P$ and $Q$, on the same side of a line $L$, the problem is to find a third point $R$ so that $PR+ RQ+RS$ is minimal, where $S$ is the unique point on $L$ such that $RS$ is perpendicular to $L.$ Consider all cases.

2005 Sharygin Geometry Olympiad, 3
Tags: geometry , locus , midpoint , chord , equal circles
Given a circle and a point $K$ inside it. An arbitrary circle equal to the given one and passing through the point $K$ has a common chord with the given circle. Find the geometric locus of the midpoints of these chords.

2003 Federal Math Competition of S&M, Problem 4
Tags: locus , geometry
An acute angle with the vertex $O$ and the rays $Op_1$ and $Op_2$ is given in a plane. Let $k_1$ be a circle with the center on $Op_1$ which is tangent to $Op_2$. Let $k_2$ be the circle that is tangent to both rays $Op_1$ and $Op_2$ and to the circle $k_1$ from outside. Find the locus of tangency points of $k_1$ and $k_2$ when center of $k_1$ moves along the ray $Op_1$.

2009 Mathcenter Contest, 2
Tags: sq , locus , geometry
Find the locus of points $P$ in the plane of a square $ABCD$ such that $$\max\{ PA,\ PC\}=\frac12(PB+PD).$$ [i](Anonymous314)[/i]

1977 Czech and Slovak Olympiad III A, 3
Tags: complex number , locus , geometry
Consider any complex units $Z,W$ with $\text{Im}\ Z\ge0,\text{Re}\,W\ge 0.$ Determine and draw the locus of all possible sums $S=Z+W$ in the complex plane.

1973 Spain Mathematical Olympiad, 6
Tags: equilateral , locus , geometry
An equilateral triangle of altitude $1$ is considered. For every point $P$ on the interior of the triangle, denote by $x, y , z$ the distances from the point $P$ to the sides of the triangle. a) Prove that for every point $P$ inside the triangle it is true that $x + y + z = 1$. b) For which points of the triangle does it hold that the distance to one side is greater than the sum of the distances to the other two? c) We have a bar of length $1$ and we break it into three pieces. find the probability that with these pieces a triangle can be formed.

1974 Chisinau City MO, 83
Tags: centroid , equilateral , locus , geometry
Let $O$ be the center of the regular triangle $ABC$. Find the set of all points $M$ such that any line containing the point $M$ intersects one of the segments $AB, OC$.

Estonia Open Senior - geometry, 1995.2.4
Tags: locus , sum , geometry , distance
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 $4$.

1984 Bulgaria National Olympiad, Problem 6
Tags: 3d geometry , locus , parallelogram , pyramid , geometry
Let there be given a pyramid $SABCD$ whose base $ABCD$ is a parallelogram. Let $N$ be the midpoint of $BC$. A plane $\lambda$ intersects the lines $SC,SA,AB$ at points $P,Q,R$ respectively such that $\overline{CP}/\overline{CS}=\overline{SQ}/\overline{SA}=\overline{AR}/\overline{AB}$. A point $M$ on the line $SD$ is such that the line $MN$ is parallel to $\lambda$. Show that the locus of points $M$, when $\lambda$ takes all possible positions, is a segment of the length $\frac{\sqrt5}2SD$.

1988 Greece National Olympiad, 2
Tags: locus , vector , geometry , analytic geometry
Given regular $1987$ -gon on plane with vertices $A_1, A_2,..., A_{1987}$. Find locus of points M of the plane sych that $$\left|\overrightarrow{MA_1}+\overrightarrow{MA_2}+...+\overrightarrow{MA_{1987}}\right| \le 1987$$.

1935 Moscow Mathematical Olympiad, 014
Tags: cube , locus in space , locus , geometry
Find the locus of points on the surface of a cube that serve as the vertex of the smallest angle that subtends the diagonal.

1989 Swedish Mathematical Competition, 4
Tags: 3d geometry , locus , sphere , tetrahedron , geometry
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 $AP$.

1949-56 Chisinau City MO, 45
Tags: locus , tangent , circles , equal segments
Determine the locus of points, from which the tangent segments to two given circles are equal.

2007 Sharygin Geometry Olympiad, 4
Tags: locus , orthocenter , locus problems , geometry
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 Greece JBMO TST, 5
Tags: area of triangle , locus , locus problems , symmetry , area , geometry
$\Phi$ 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 $\Phi$.

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

2018 Peru Cono Sur TST, 7
Tags: geometry , locus
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 $AD$.

2018 Oral Moscow Geometry Olympiad, 5
Tags: geometry , locus , orthocenter , midpoint , fixed
The circle circumscribed about an acute triangle $ABC$ and the vertex $C$ are fixed. Orthocenter $H$ moves in a circle with center at point $C$. Find the locus of the midpoints of the segments connecting the feet of altitudes drawn from vertices $A$ and $B$.

2010 Laurențiu Panaitopol, Tulcea, 3
Tags: invariant , locus , circumcircle , geometry
Let $ R $ be the circumradius of a triangle $ ABC. $ The points $ B,C, $ lie on a circle of radius $ \rho $ that intersects $ AB,AC $ at $ E,D, $ respectively. $ \rho' $ is the circumradius of $ ADE. $ Show that there exists a triangle with sides $ R,\rho ,\rho' , $ and having an angle whose value doesn't depend on $ \rho . $ [i]Laurențiu Panaitopol[/i]

2007 Sharygin Geometry Olympiad, 4
Tags: locus , locus problems , midpoint , geometry , circumcircle
Given a triangle $ABC$. An arbitrary point $P$ is chosen on the circumcircle of triangle $ABH$ ($H$ is the orthocenter of triangle $ABC$). Lines $AP$ and $BP$ meet the opposite sidelines of the triangle at points $A' $ and $B'$, respectively. Determine the locus of midpoints of segments $A'B'$.

2024 ITAMO, 2
Tags: locus , geometry
We are given a unit square in the plane. A point $M$ in the plane is called [i]median [/i]if there exists points $P$ and $Q$ on the boundary of the square such that $PQ$ has length one and $M$ is the midpoint of $PQ$. Determine the geometric locus of all median points.

1998 Mexico National Olympiad, 2
Tags: geometry , locus , circles
Rays $l$ and $m$ forming an angle of $a$ are drawn from the same point. Let $P$ be a fixed point on $l$. For each circle $C$ tangent to $l$ at $P$ and intersecting $m$ at $Q$ and $R$, let $T$ be the intersection point of the bisector of angle $QPR$ with $C$. Describe the locus of $T$ and justify your answer.

1973 IMO Shortlist, 2
Tags: geometry , locus problems , locus , parallelogram , circles
Given a circle $K$, find the locus of vertices $A$ of parallelograms $ABCD$ with diagonals $AC \leq BD$, such that $BD$ is inside $K$.

IV Soros Olympiad 1997 - 98 (Russia), 9.5
Tags: rotation , geometry , locus , geometric transformation
Given triangle $ABC$. Find the locus of points $M$ such that there is a rotation with center at $M$ that takes $C$ to a certain point on side $AB$.

2017 Sharygin Geometry Olympiad, P5
Tags: locus , acute triangle , incenter , geometry
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.