Found problems: 68
1966 IMO Longlists, 16
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 $Q.$
2009 Romania National Olympiad, 1
On the sides $ AB,AC $ of a triangle $ ABC, $ consider the points $ M, $ respectively, $ N $ such that $ M\neq A\neq N $ and $ \frac{MB}{MA}\neq\frac{NC}{NA}. $ Show that the line $ MN $ passes through a point not dependent on $ M $ and $ N. $
2017 Vietnamese Southern Summer School contest, Problem 4
Let $ABC$ be a triangle. A point $P$ varies inside $BC$. Let $Q, R$ be the points on $AC, AB$ in that order, such that $PQ\parallel AB, PR\parallel AC$.
1. Prove that, when $P$ varies, the circumcircle of triangle $AQR$ always passes through a fixed point $X$ other than $A$.
2. Extend $AX$ so that it cuts the circumcircle of $ABC$ a second time at point $K$. Prove that $AX=XK$.
1969 IMO Shortlist, 53
$(POL 2)$ Given two segments $AB$ and $CD$ not in the same plane, find the locus of points $M$ such that $MA^2 +MB^2 = MC^2 +MD^2.$
2004 Germany Team Selection Test, 2
Let $d$ be a diameter of a circle $k$, and let $A$ be an arbitrary point on this diameter $d$ in the interior of $k$. Further, let $P$ be a point in the exterior of $k$. The circle with diameter $PA$ meets the circle $k$ at the points $M$ and $N$.
Find all points $B$ on the diameter $d$ in the interior of $k$ such that
\[\measuredangle MPA = \measuredangle BPN \quad \text{and} \quad PA \leq PB.\]
(i. e. give an explicit description of these points without using the points $M$ and $N$).
2017 Yasinsky Geometry Olympiad, 6
Given a circle $\omega$ of radius $r$ and a point $A$, which is far from the center of the circle at a distance $d<r$. Find the geometric locus of vertices $C$ of all possible $ABCD$ rectangles, where points $B$ and $D$ lie on the circle $\omega$.
1978 IMO, 2
We consider a fixed point $P$ in the interior of a fixed sphere$.$ We construct three segments $PA, PB,PC$, perpendicular two by two$,$ with the vertexes $A, B, C$ on the sphere$.$ We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as edges$.$ Find the locus of the point $Q$ when $A, B, C$ take all the positions compatible with our problem.
1995 China Team Selection Test, 2
Given a fixed acute angle $\theta$ and a pair of internally tangent circles, let the line $l$ which passes through the point of tangency, $A$, cut the larger circle again at $B$ ($l$ does not pass through the centers of the circles). Let $M$ be a point on the major arc $AB$ of the larger circle, $N$ the point where $AM$ intersects the smaller circle, and $P$ the point on ray $MB$ such that $\angle MPN = \theta$. Find the locus of $P$ as $M$ moves on major arc $AB$ of the larger circle.
2006 Sharygin Geometry Olympiad, 13
Two straight lines $a$ and $b$ are given and also points $A$ and $B$. Point $X$ slides along the line $a$, and point $Y$ slides along the line $b$, so that $AX \parallel BY$. Find the locus of the intersection point of $AY$ with $XB$.
1999 Greece JBMO TST, 5
$\Phi$ 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 $\Phi$.
1966 IMO Longlists, 55
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 $[40^\circ, 70^\circ].$
2010 Sharygin Geometry Olympiad, 6
An arbitrary line passing through vertex $B$ of triangle $ABC$ meets side $AC$ at point $K$ and the circumcircle in point $M$. Find the locus of circumcenters of triangles $AMK$.
2007 Sharygin Geometry Olympiad, 4
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 $A'B'$.
1992 IMO Longlists, 17
In the plane let $\,C\,$ be a circle, $\,L\,$ a line tangent to the circle $\,C,\,$ and $\,M\,$ a point on $\,L$. Find the locus of all points $\,P\,$ with the following property: there exists two points $\,Q,R\,$ on $\,L\,$ such that $\,M\,$ is the midpoint of $\,QR\,$ and $\,C\,$ is the inscribed circle of triangle $\,PQR$.
1960 IMO Shortlist, 5
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 $ZY=2XZ$.
1960 IMO Shortlist, 7
An isosceles trapezoid with bases $a$ and $c$ and altitude $h$ is given.
a) On the axis of symmetry of this trapezoid, find all points $P$ such that both legs of the trapezoid subtend right angles at $P$;
b) Calculate the distance of $p$ from either base;
c) Determine under what conditions such points $P$ actually exist. Discuss various cases that might arise.
1992 IMO Shortlist, 20
In the plane let $\,C\,$ be a circle, $\,L\,$ a line tangent to the circle $\,C,\,$ and $\,M\,$ a point on $\,L$. Find the locus of all points $\,P\,$ with the following property: there exists two points $\,Q,R\,$ on $\,L\,$ such that $\,M\,$ is the midpoint of $\,QR\,$ and $\,C\,$ is the inscribed circle of triangle $\,PQR$.
1978 IMO Longlists, 46
We consider a fixed point $P$ in the interior of a fixed sphere$.$ We construct three segments $PA, PB,PC$, perpendicular two by two$,$ with the vertexes $A, B, C$ on the sphere$.$ We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as edges$.$ Find the locus of the point $Q$ when $A, B, C$ take all the positions compatible with our problem.