Olympiad problems

1969 Spain Mathematical Olympiad, 1

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 $\gamma$ .

1963 German National Olympiad, 5

Tags: geometry , Locus , midpoint , square

Given is a square with side length $a$. A distance $PQ$ of length $p$, where $p < a$, moves so that its end points are always on the sides of the square. What is the geometric locus of the midpoints of the segments $PQ$?

1985 Czech And Slovak Olympiad IIIA, 4

Tags: geometry , Locus

Two straight lines $p, q$ are given in the plane and on the straight line $q$ there is a point $F$, $F \not\in p$. Determine the set of all points $X$ that can be obtained by this construction: In the plane we choose a point $S$ that lies neither on $p$ nor on $q$, and we construct a circle $k$ with center $S$ that is tangent to the line $p$. On the circle $k$ we choose a point $T$ such that so that $ST \parallel q$. If the line $FT$ intersects the line $p$ at the point $U$, $X$ is the intersection of the lines $SU$ and $q$

2007 Hanoi Open Mathematics Competitions, 13

Tags: geometry , areas , Locus

Let be given triangle $ABC$. Find all points $M$ such that area of $\vartriangle MAB$= area of $\vartriangle MAC$

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.

2006 Hanoi Open Mathematics Competitions, 6

Tags: geometry , hexagon , Locus

The figure $ABCDEF$ is a regular hexagon. Find all points $M$ belonging to the hexagon such that Area of triangle $MAC =$ Area of triangle $MCD$.

1974 Czech and Slovak Olympiad III A, 6

Tags: geometry , geometric transformation , rotation , Locus , areas , square

Let a unit square $\mathcal D$ be given in the plane. For any point $X$ in the plane denote $\mathcal D_X$ the image of $\mathcal D$ in rotation with respect to origin $X$ by $+90^\circ.$ Find the locus of all $X$ such that the area of union $\mathcal D\cup\mathcal D_X$ is at most 1.5.

1962 Czech and Slovak Olympiad III A, 3

Tags: geometry , 3D geometry , lines , circle , Locus

Let skew lines $PM, QN$ be given such that $PM\perp PQ\perp QN$. Let a plane $\sigma\perp PQ$ containing the midpoint $O$ of segment $PQ$ be given and in it a circle $k$ with center $O$ and given radius $r$. Consider all segments $XY$ with endpoint $X, Y$ on lines $PM, QN$, respectively, which contain a point of $k$. Show that segments $XY$ have the same length. Find the locus of all such points $X$.

1986 IMO Longlists, 47

Tags: geometry , rotation , polygon , Locus , Locus problems , IMO , IMO 1986

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 $X$.

Champions Tournament Seniors - geometry, 2007.3

Tags: geometry , circumcircle , Locus , equal segments , Champions Tournament

Given a triangle $ABC$. Point $M$ moves along the side $BA$ and point $N$ moves along the side $AC$ beyond point $C$ such that $BM=CN$. Find the geometric locus of the centers of the circles circumscribed around the triangle $AMN$.

1963 All Russian Mathematical Olympiad, 040

Tags: geometry , isosceles , Locus

Given an isosceles triangle. Find the set of the points inside the triangle such, that the distance from that point to the base equals to the geometric mean of the distances to the sides.

1982 Austrian-Polish Competition, 2

Tags: Locus , Tangents , circle , convex , geometry

Let $F$ be a closed convex region inside a circle $C$ with center $O$ and radius $1$. Furthermore, assume that from each point of $C$ one can draw two rays tangent to $F$ which form an angle of $60^o$. Prove that $F$ is the disc centered at $O$ with radius $1/2$.

1942 Putnam, A1

Tags: Putnam , square , Locus

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.

1986 IMO, 1

Tags: geometry , rotation , polygon , Locus , Locus problems , IMO , IMO 1986

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

Tags: geometry , Locus , square , Equilateral , Triangle

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 $M.$

2001 Taiwan National Olympiad, 4

Tags: geometry , similar triangles , incenter , Locus , circumcircle

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 $Q$.

1964 Bulgaria National Olympiad, Problem 3

Tags: geometry , Locus

There are given two intersecting lines $g_1,g_2$ and a point $P$ in their plane such that $\angle(g1,g2)\ne90^\circ$. Its symmetrical points on any point $M$ in the same plane with respect to the given lines are $M_1$ and $M_2$. Prove that: (a) the locus of the point $M$ for which the points $M_1,M_2$ and $P$ lie on a common line is a circle $k$ passing through the intersection point of $g_1$ and $g_2$. (b) the point $P$ is an orthocenter of a triangle, inscribed in the circle $k$ whose sides lie at the lines $g_1$ and $g_2$.

2007 Sharygin Geometry Olympiad, 10

Tags: geometry , Locus , Equilateral , Equilateral Triangle , Locus problems

Find the locus of centers of regular triangles such that three given points $A, B, C$ lie respectively on three lines containing sides of the triangle.

1986 IMO Shortlist, 16

Tags: geometry , rotation , polygon , Locus , Locus problems , IMO , IMO 1986

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 $X$.

1949-56 Chisinau City MO, 44

Tags: geometry , Locus , Difference

Determine the locus of points, for each of which the difference between the squares of the distances to two given points is a constant value.

1956 Moscow Mathematical Olympiad, 325

Tags: geometry , equal segments , midpoint , Locus

On sides $AB$ and $CB$ of $\vartriangle ABC$ there are drawn equal segments, $AD$ and $CE$, respectively, of arbitrary length (but shorter than min($AB,BC$)). Find the locus of midpoints of all possible segments $DE$.

1966 IMO Shortlist, 55

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

Given the vertex $A$ and the centroid $M$ of a triangle $ABC$, find the locus of vertices $B$ such that all the angles of the triangle lie in the interval $[40^\circ, 70^\circ].$

1986 IMO Longlists, 33

Tags: geometry , rotation , polygon , Locus , Locus problems , IMO , IMO 1986

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 $X$.

II Soros Olympiad 1995 - 96 (Russia), 9.9

Tags: geometry , Locus

Two points $A$ and $B$ are given on the plane. An arbitrary circle passes through $B$ and intersects the straight line $AB$ for second time at a point $K$, different from $A$. A circle passing through $A$, $K$ and the center of the first circle intersects the first one for second time at point $M$. Find the locus of points $M$.

1978 Bulgaria National Olympiad, Problem 2

Tags: circles , Locus , geometry

$k_1$ denotes one of the arcs formed by intersection of the circumference $k$ and the chord $AB$. $C$ is the middle point of $k_1$. On the half line (ray) $PC$ is drawn the segment $PM$. Find the locus formed from the point $M$ when $P$ is moving on $k_1$. [i]G. Ganchev[/i]