Olympiad problems

Kyiv City MO 1984-93 - geometry, 1992.8.3

Find the locus of the intersection points of the medians all triangles inscribed in a given circle.

1963 IMO, 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 $BC$.

1966 IMO Longlists, 16

Tags: geometry , locus problems , locus , circles

We are given a circle $K$ with center $S$ and radius $1$ and a square $Q$ with center $M$ and side $2$. Let $XY$ be the hypotenuse of an isosceles right triangle $XY Z$. Describe the locus of points $Z$ as $X$ varies along $K$ and $Y$ varies along the boundary of $Q.$

2004 Nicolae Păun, 3

Tags: geometry , 3d geometry , sphere , cube , locus

[b]a)[/b] Show that the sum of the squares of the minimum distances from a point that is situated on a sphere to the faces of the cube that circumscribe the sphere doesn't depend on the point. [b]b)[/b] Show that the sum of the cubes of the minimum distances from a point that is situated on a sphere to the faces of the cube that circumscribe the sphere doesn't depend on the point. [i]Alexandru Sergiu Alamă[/i]

1942 Putnam, A1

Tags: 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.

2003 Federal Math Competition of S&M, Problem 4

Tags: geometry , locus

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 $Op_1$.

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.

Ukrainian TYM Qualifying - geometry, 2013.6

Tags: geometry , locus

Given a circle $\omega$, on which marks the points $A,B,C$. Let $BF$ and $CE$ be the altitudes of the triangle $ABC$, $M$ be the midpoint of the side $AC$. Find a the locus of the intersection points of the lines $BF$ and E$M$ for all positions of point $A$ , as $A$ moves on $\omega$.

1949-56 Chisinau City MO, 31

Tags: geometry , locus , chord , midpoint

Find the locus of the points that are the midpoints of the chords of the secant to the given circle and passing through a given point.

1970 Czech and Slovak Olympiad III A, 5

Tags: line , geometry , geometric transformation , rotation , homothety , locus

Let a real number $k$ and points $S,A,SA=1$ in plane be given. Denote $A'$ the image of $A$ under rotation by an oriented angle $\varphi$ with respect to center $S$. Similarly, let $A''$ be the image of $A'$ under homothety with the factor $\frac{1}{\cos\varphi-k\sin\varphi}$ with respect to center $S.$ Denote the locus \[\ell=\bigl\{A''\mid\varphi\in(-\pi,\pi],\cos\varphi-k\sin\varphi\neq0\bigr\}.\] Show that $\ell$ is a line containing $A.$

1973 IMO Shortlist, 1

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

Let a tetrahedron $ABCD$ be inscribed in a sphere $S$. Find the locus of points $P$ inside the sphere $S$ for which the equality \[\frac{AP}{PA_1}+\frac{BP}{PB_1}+\frac{CP}{PC_1}+\frac{DP}{PD_1}=4\] holds, where $A_1,B_1, C_1$, and $D_1$ are the intersection points of $S$ with the lines $AP,BP,CP$, and $DP$, respectively.

2012 Tournament of Towns, 4

Tags: geometry , cyclic , locus , circles , tangent circle

A quadrilateral $ABCD$ with no parallel sides is inscribed in a circle. Two circles, one passing through $A$ and $B$, and the other through $C$ and $D$, are tangent to each other at $X$. Prove that the locus of $X$ is a circle.

2006 Sharygin Geometry Olympiad, 22

Tags: geometry , fixed , circumcenter , locus

Given points $A, B$ on a circle and a point $P$ not lying on the circle. $X$ is an arbitrary point of the circle, $Y$ is the intersection point of lines $AX$ and $BP$. Find the locus of the centers of the circles circumscribed around the triangles $PXY$.