Olympiad problems

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 $.

2012 Czech-Polish-Slovak Junior Match, 2

Tags: midpoint , parallel , geometry , Locus , arc

On the circle $k$, the points $A,B$ are given, while $AB$ is not the diameter of the circle $k$. Point $C$ moves along the long arc $AB$ of circle $k$ so that the triangle $ABC$ is acute. Let $D,E$ be the feet of the altitudes from $A, B$ respectively. Let $F$ be the projection of point $D$ on line $AC$ and $G$ be the projection of point $E$ on line $BC$. (a) Prove that the lines $AB$ and $FG$ are parallel. (b) Determine the set of midpoints $S$ of segment $FG$ while along all allowable positions of point $C$.

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 $\beta$.

2008 Switzerland - Final Round, 5

Tags: geometry , Locus , square

Let $ABCD$ be a square with side length $1$. Find the locus of all points $P$ with the property $AP\cdot CP + BP\cdot DP = 1$.

1984 Bulgaria National Olympiad, Problem 6

Tags: geometry , 3D geometry , pyramid , parallelogram , Locus

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 $\frac{\sqrt5}2SD$.

1956 Moscow Mathematical Olympiad, 339

Tags: geometry , Locus , projection

Find the union of all projections of a given line segment $AB$ to all lines passing through a given point $O$.

2016 Sharygin Geometry Olympiad, 3

Tags: geometry , inradius , Locus , circumcircle

Assume that the two triangles $ABC$ and $A'B'C'$ have the common incircle and the common circumcircle. Let a point $P$ lie inside both the triangles. Prove that the sum of the distances from $P$ to the sidelines of triangle $ABC$ is equal to the sum of distances from $P$ to the sidelines of triangle $A'B'C'$.

2000 Tournament Of Towns, 3

Tags: geometry , rectangle , Locus , circle , fixed

$A$ is a fixed point inside a given circle. Determine the locus of points $C$ such that $ABCD$ is a rectangle with $B$ and $D$ on the circumference of the given circle. (M Panov)

1960 IMO, 5

Tags: geometry , 3D geometry , cube , Locus , Locus problems , IMO , IMO 1960

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 $ZY=2XZ$.

1953 Putnam, B6

Tags: Putnam , Locus , circle , minima

Let $P$ and $Q$ be any points inside a circle $C$ with center $O$ such that $OP=OQ.$ Determine the location of a point $Z$ on $C$ such that $PZ+QZ$ is minimal.

1957 Moscow Mathematical Olympiad, 351

Tags: geometry , Locus , rectangle , concentric circles

Given two concentric circles and a pair of parallel lines. Find the locus of the fourth vertices of all rectangles with three vertices on the concentric circles, two vertices on one circle and the third on the other and with sides parallel to the given lines.

2014 Contests, 1

Tags: trigonometry , geometry , orthocenter , Locus

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]

1963 IMO Shortlist, 2

Tags: geometry , 3D geometry , sphere , Locus , Locus problems , IMO , IMO 1963

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 $BC$.

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.

1989 Swedish Mathematical Competition, 4

Tags: geometry , 3D geometry , tetrahedron , Locus , sphere

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 $AP$.

Estonia Open Senior - geometry, 2019.2.5

Tags: geometry , Locus , Equilateral

The plane has a circle $\omega$ and a point $A$ outside it. For any point $C$, the point $B$ on the circle $\omega$ is defined such that $ABC$ is an equilateral triangle with vertices $A, B$ and $C$ listed clockwise. Prove that if point $B$ moves along the circle $\omega$, then point $C$ also moves along a circle.

1969 IMO Longlists, 39

Tags: geometry , 3D geometry , cube , Locus , Locus problems , IMO Shortlist , IMO Longlist

$(HUN 6)$ 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.

1946 Moscow Mathematical Olympiad, 122

Tags: Sum , areas , Locus , ratio , geometry

On the sides $PQ, QR, RP$ of $\vartriangle PQR$ segments $AB, CD, EF$ are drawn. Given a point $S_0$ inside triangle $\vartriangle PQR$, find the locus of points $S$ for which the sum of the areas of triangles $\vartriangle SAB$, $\vartriangle SCD$ and $\vartriangle SEF$ is equal to the sum of the areas of triangles $\vartriangle S_0AB$, $\vartriangle S_0CD$, $\vartriangle S0EF$. Consider separately the case $$\frac{AB}{PQ }= \frac{CD}{QR} = \frac{EF}{RP}.$$

1967 IMO Shortlist, 3

Tags: geometry , incenter , angle bisector , Locus , Locus problems , IMO Shortlist , IMO Longlist

Circle $k$ and its diameter $AB$ are given. Find the locus of the centers of circles inscribed in the triangles having one vertex on $AB$ and two other vertices on $k.$

1941 Moscow Mathematical Olympiad, 089

Tags: geometry , skew , Locus

Given two skew perpendicular lines in space, find the set of the midpoints of all segments of given length with the endpoints on these lines.

1966 IMO Shortlist, 28

Tags: geometry , circle , Locus , Locus problems , IMO Longlist , IMO Shortlist

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.

1954 Czech and Slovak Olympiad III A, 4

Tags: geometry , 3D geometry , ratio , Locus , cube

Consider a cube $ABCDA'B'C'D$ (with $AB\perp AA'\parallel BB'\parallel CC'\parallel DD$). Let $X$ be an inner point of the segment $AB$ and denote $Y$ the intersection of the edge $AD$ and the plane $B'D'X$. (a) Let $M=B'Y\cap D'X$. Find the locus of all $M$s. (b) Determine whether there is a quadrilateral $B'D'YX$ such that its diagonals divide each other in the ratio 1:2.