Olympiad problems

1953 Polish MO Finals, 2

Find the geometric locus of the center of a rectangle whose vertices lie on the perimeter of a given triangle.

2011 Sharygin Geometry Olympiad, 13

Tags: tetrahedron , locus , centroid , inscribed triangle , 3d geometry , geometry

a) Find the locus of centroids for triangles whose vertices lie on the sides of a given triangle (each side contains a single vertex). b) Find the locus of centroids for tetrahedrons whose vertices lie on the faces of a given tetrahedron (each face contains a single vertex).

1962 Czech and Slovak Olympiad III A, 3

Tags: geometry , 3d geometry , line , locus , circles

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 $X$.

2009 Federal Competition For Advanced Students, P2, 3

Tags: locus , congruent triangles , geometry

Let $P$ be the point in the interior of $\vartriangle ABC$. Let $D$ be the intersection of the lines $AP$ and $BC$ and let $A'$ be the point such that $\overrightarrow{AD} = \overrightarrow{DA'}$. The points $B'$ and $C'$ are defined in the similar way. Determine all points $P$ for which the triangles $A'BC, AB'C$, and $ABC'$ are congruent to $\vartriangle ABC$.

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 $O$.

2017 Ukrainian Geometry Olympiad, 3

Tags: fixed , intersect , locus , locus problems , circles , geometry

Circles ${w}_{1},{w}_{2}$ intersect at points ${{A}_{1}} $ and ${{A}_{2}} $. Let $B$ be an arbitrary point on the circle ${{w}_{1}}$, and line $B{{A}_{2}}$ intersects circle ${{w}_{2}}$ at point $C$. Let $H$ be the orthocenter of $\Delta B{{A}_{1}}C$. Prove that for arbitrary choice of point $B$, the point $H$ lies on a certain fixed circle.

2023 Sharygin Geometry Olympiad, 19

Tags: geometry , locus , circle intersections

A cyclic quadrilateral $ABCD$ is given. An arbitrary circle passing through $C$ and $D$ meets $AC,BC$ at points $X,Y$ respectively. Find the locus of common points of circles $CAY$ and $CBX$.

1979 Bundeswettbewerb Mathematik, 2

Tags: geometry , locus , circles , incenter

A circle $k$ with center $M$ and radius $r$ is given. Find the locus of the incenters of all obtuse-angled triangles inscribed in $k$.

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.

2006 Sharygin Geometry Olympiad, 10.3

Tags: geometry , tangent , tangent circle , locus

Given a circle and a point $P$ inside it, different from the center. We consider pairs of circles tangent to the given internally and to each other at point $P$. Find the locus of the points of intersection of the common external tangents to these circles.

1986 IMO Longlists, 47

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 $X$.

2002 Moldova Team Selection Test, 3

Tags: geometry , locus , minimum

A triangle $ABC$ is inscribed in a circle $G$. For any point $M$ inside the triangle, $A_1$ denotes the intersection of the ray $AM$ with $G$. Find the locus of point $M$ for which $\frac{BM\cdot CM}{MA_1}$ is minimal, and find this minimum value.

1969 IMO Longlists, 53

Tags: trigonometry , locus , locus problems , geometry

$(POL 2)$ Given two segments $AB$ and $CD$ not in the same plane, find the locus of points $M$ such that $MA^2 +MB^2 = MC^2 +MD^2.$

1965 IMO, 5

Tags: parallelogram , locus , locus problems , geometry

Consider $\triangle OAB$ with acute angle $AOB$. Thorugh a point $M \neq O$ perpendiculars are drawn to $OA$ and $OB$, the feet of which are $P$ and $Q$ respectively. The point of intersection of the altitudes of $\triangle OPQ$ is $H$. What is the locus of $H$ if $M$ is permitted to range over a) the side $AB$; b) the interior of $\triangle OAB$.

1963 German National Olympiad, 5

Tags: midpoint , locus , square , geometry

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 $PQ$?

V Soros Olympiad 1998 - 99 (Russia), 10.5

Tags: locus , geometry

An isosceles triangle $ABC$ ($AB = BC$) is given on the plane. Find the locus of points $M$ of the plane such that $ABCM$ is a convex quadrilateral and $\angle MAC + \angle CMB = 90^o$.

2000 Regional Competition For Advanced Students, 3

Tags: geometry , midpoint , locus

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 $c$?

1986 Bulgaria National Olympiad, Problem 5

Tags: circles , geometry , locus

Let $A$ be a fixed point on a circle $k$. Let $B$ be any point on $k$ and $M$ be a point such that $AM:AB=m$ and $\angle BAM=\alpha$, where $m$ and $\alpha$ are given. Find the locus of point $M$ when $B$ describes the circle $k$.

2006 Sharygin Geometry Olympiad, 14

Tags: geometry , diameter , locus , locus problems , orthocenter

Given a circle and a fixed point $P$ not lying on it. Find the geometrical locus of the orthocenters of the triangles $ABP$, where $AB$ is the diameter of the circle.

1974 Czech and Slovak Olympiad III A, 6

Tags: rotation , geometry , square , locus , area , geometric transformation

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.

2007 Nicolae Coculescu, 4

Tags: analytic geometry , locus , infimum , geometry

Let $ M $ be a point in the interior of a triangle $ ABC, $ let $ D $ be the intersection of $ AM $ with $ BC, $ let $ E $ be the intersection of $ M $ with AC, let $ F $ be the intersection of $ CM $ with $ AB. $ Knowing that the expression $$ \frac{MA}{MD}\cdot \frac{MB}{ME}\cdot \frac{MC}{MF} $$ is minimized, describe the point $ M. $

1949-56 Chisinau City MO, 44

Tags: locus , difference , geometry

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.

1957 Kurschak Competition, 1

Tags: geometry , 3d geometry , tetrahedron , locus

$ABC$ is an acute-angled triangle. $D$ is a variable point in space such that all faces of the tetrahedron $ABCD$ are acute-angled. $P$ is the foot of the perpendicular from $D$ to the plane $ABC$. Find the locus of $P$ as $D$ varies.

1982 Spain Mathematical Olympiad, 8

Tags: geometry , locus , analytic geometry

Given a set $C$ of points in the plane, it is called the distance of a point $P$ from the plane to the set $C$ at the smallest of the distances from $P$ to each of the points of $C$. Let the sets be $C = \{A,B\}$, with $A = (1, 0)$ and $B = (2, 0)$; and $C'= \{A',B'\}$ with $A' = (0, 1)$ and $B' = (0, 7)$, in an orthogonal reference system. Find and draw the set $M$ of points in the plane that are equidistant from $C$ and $C'$ . Study whether the function whose graph is the set $M$ previously obtained is derivable.

1969 IMO Shortlist, 12

Tags: geometry , 3d geometry , tetrahedron , locus , locus problems

$(CZS 1)$ Given a unit cube, find the locus of the centroids of all tetrahedra whose vertices lie on the sides of the cube.