Olympiad problems

1949-56 Chisinau City MO, 32

Determine the locus of points that are the midpoints of segments of equal length, the ends of which lie on the sides of a given right angle.

1924 Eotvos Mathematical Competition, 2

Tags: geometry , Locus , fixed

If $O$ is a given point, $\ell$ a given line, and $a$ a given positive number, find the locus of points $P$ for which the sum of the distances from $P$ to $O$ and from $P$ to $\ell$ is $a$.

1962 All Russian Mathematical Olympiad, 014

Tags: geometry , Locus

Given the circumference $s$ and the straight line $l$, passing through the centre $O$ of $s$. Another circumference $s'$ passes through the point $O$ and has its centre on the $l$. Describe the set of the points $M$, where the common tangent of $s$ and $s'$ touches $s'$.

2021/2022 Tournament of Towns, P4

Tags: geometry , 3D geometry , Locus , Tournament of Towns

Given is a segment $AB$. Three points $X, Y, Z$ are picked in the space so that $ABX$ is an equilateral triangle and $ABYZ$ is a square. Prove that the orthocenters of all triangles $XYZ$ obtained in this way belong to a fixed circle. [i]Alexandr Matveev[/i]

1947 Moscow Mathematical Olympiad, 132

Tags: geometry , Locus , equidistant , distance

Given line $AB$ and point $M$. Find all lines in space passing through $M$ at distance $d$.

1966 IMO Longlists, 16

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

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

Tags: geometry , Locus problems , Locus , vectorial geometry

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. $

1963 Dutch Mathematical Olympiad, 2

Tags: geometry , Locus , geometric inequality , midpoint

The straight lines $k$ and $\ell$ intersect at right angles. A line intersects $k$ in $A$ and $\ell$ in $B$. Consider all straight line segments $PQ$ ($P$ on $k$ and $Q$ on $\ell$), which makes an angle of $45^o$ with $AB$. (a) Determine the locus of the midpoints of the line segments $PQ$, (b) If the perpendicular bisector of such a line segment $PQ$ intersects the line $k$ at $K$ and the line $\ell$ at $L$, then prove that $KL \ge PQ$. [hide=original wording of second sentence]De loodrechte snijlijn van k en l snijdt k in A en t in B[/hide]

1971 Czech and Slovak Olympiad III A, 5

Tags: geometry , Equilateral , Triangles , Locus

Let $ABC$ be a given triangle. Find the locus $\mathbf M$ of all vertices $Z$ such that triangle $XYZ$ is equilateral where $X$ is any point of segment $AB$ and $Y\neq X$ lies on ray $AC.$

2017 Vietnamese Southern Summer School contest, Problem 4

Tags: geometry , Locus , Locus problems

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 $AX=XK$.

2004 Tournament Of Towns, 4

Tags: geometry , Locus , Circumcenter , circumcircle

A circle with the center $I$ is entirely inside of a circle with center $O$. Consider all possible chords $AB$ of the larger circle which are tangent to the smaller one. Find the locus of the centers of the circles circumscribed about the triangle $AIB$.

1969 IMO Shortlist, 53

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

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

2010 Sharygin Geometry Olympiad, 2

Tags: geometry , Locus , circle

Two points $A$ and $B$ are given. Find the locus of points $C$ such that triangle $ABC$ can be covered by a circle with radius $1$. (Arseny Akopyan)

1977 Bulgaria National Olympiad, Problem 4

Tags: Locus , geometry

Vertices $A$ and $C$ of the quadrilateral $ABCD$ are fixed points of the circle $k$ and each of the vertices $B$ and $D$ is moving to one of the arcs of $k$ with ends $A$ and $C$ in such a way that $BC=CD$. Let $M$ be the intersection point of $AC$ and $BD$ and $F$ is the center of the circumscribed circle around $\triangle ABM$. Prove that the locus of $F$ is an arc of a circle. [i]J. Tabov[/i]

1955 Moscow Mathematical Olympiad, 293

Tags: Locus , equal segments , geometry

Consider a quadrilateral $ABCD$ and points $K, L, M, N$ on sides $AB, BC, CD$ and $AD$, respectively, such that $KB = BL = a, MD = DN = b$ and $KL \nparallel MN$. Find the set of all the intersection points of $KL$ with $MN$ as $a$ and $b$ vary.

2007 Hanoi Open Mathematics Competitions, 8

Tags: geometry , Locus , Equilateral

Let $ABC$ be an equilateral triangle. For a point $M$ inside $\vartriangle ABC$, let $D,E,F$ be the feet of the perpendiculars from $M$ onto $BC,CA,AB$, respectively. Find the locus of all such points $M$ for which $\angle FDE$ is a right angle.

1984 Spain Mathematical Olympiad, 5

Tags: geometry , Locus , arc

Let $A$ and $A' $ be fixed points on two equal circles in the plane and let $AB$ and $A' B'$ be arcs of these circles of the same length $x$. Find the locus of the midpoint of segment $BB'$ when $x$ varies: (a) if the arcs have the same direction, (b) if the arcs have opposite directions.

1987 Tournament Of Towns, (151) 2

Tags: geometry , rhombus , Locus , angles

Find the locus of points $M$ inside the rhombus $ABCD$ such that the sum of angles $AMB$ and $CMD$ equals $180^o$ .

2005 Sharygin Geometry Olympiad, 9.3

Tags: Locus , midpoints , geometry , arc

Given a circle and points $A, B$ on it. Draw the set of midpoints of the segments, one of the ends of which lies on one of the arcs $AB$, and the other on the second.

1953 Czech and Slovak Olympiad III A, 4

Tags: 3D geometry , Locus

Consider skew lines $a,b$ and a plane $\rho$ that intersect both of the lines (but does not contain any of them). Choose such points $X\in a,Y\in b$ that $XY\parallel\rho.$ Find the locus of midpoints $M$ of all segments $XY,$ when $X$ moves along line $a$.

1969 IMO Longlists, 4

Tags: conics , geometry , Locus , IMO Shortlist , IMO Longlist

$(BEL 4)$ Let $O$ be a point on a nondegenerate conic. A right angle with vertex $O$ intersects the conic at points $A$ and $B$. Prove that the line $AB$ passes through a fixed point located on the normal to the conic through the point $O.$

1941 Putnam, A3

Tags: Putnam , Locus

A circle of radius $a$ rolls in the plane along the $x$-axis. Show that the envelope of a diameter is a cycloid, coinciding with the cycloid traced out by a point on the circumference of a circle of diameter $a$, likewise rolling in the plane along the $x$-axis.