Olympiad problems

1956 Czech and Slovak Olympiad III A, 4

Tags: Locus , locus of points , semicircle , geometry , national olympiad

Let a semicircle $AB$ be given and let $X$ be an inner point of the arc. Consider a point $Y$ on ray $XA$ such that $XY=XB$. Find the locus of all points $Y$ when $X$ moves on the arc $AB$ (excluding the endpoints).

2018 IMAR Test, 1

Tags: geometry , Locus , romania

Let $ABC$ be a triangle whose angle at $A$ is right, and let $D$ be the foot of the altitude from $A$. A variable point $M$ traces the interior of the minor arc $AB$ of the circle $ABC$. The internal bisector of the angle $DAM$ crosses $CM$ at $N$. The line through $N$ and perpendicular to $CM$ crosses the line $AD$ at $P$. Determine the locus of the point where the line $BN$ crosses the line $CP$. [i]* * *[/i]

1970 Vietnam National Olympiad, 4

Tags: geometry , Locus , construction , tangent , diameter

$AB$ and $CD$ are perpendicular diameters of a circle. $L$ is the tangent to the circle at $A$. $M$ is a variable point on the minor arc $AC$. The ray $BM, DM$ meet the line $L$ at $P$ and $Q$ respectively. Show that $AP\cdot AQ = AB\cdot PQ$. Show how to construct the point $M$ which gives$ BQ$ parallel to $DP$. If the lines $OP$ and $BQ$ meet at $N$ find the locus of $N$. The lines $BP$ and $BQ$ meet the tangent at $D$ at $P'$ and $Q'$ respectively. Find the relation between $P'$ and $Q$'. The lines $D$P and $DQ$ meet the line $BC$ at $P"$ and $Q"$ respectively. Find the relation between $P"$ and $Q"$.

VMEO III 2006, 11.2

Tags: geometry , Locus , midpoint

Let $ABCD$ be an isosceles trapezoid, with a large base $CD$ and a small base $AB$. Let $M$ be any point on side $AB$ and $(d)$ be the line through $M$ and perpendicular to $AB$. Two rays $Mx$ and $My$ are said to satisfy the condition $(T)$ if they are symmetric about each other through $(d)$ and intersect the two rays $AD$ and $BC$ at $E$ and $F$ respectively. Find the locus of the midpoint of the segment $EF$ when the two rays $Mx$ and $My$ change and satisfy condition $(T)$.

2014 Sharygin Geometry Olympiad, 13

Tags: geometry , Locus , Circumcenter

Let $AC$ be a fixed chord of a circle $\omega$ with center $O$. Point $B$ moves along the arc $AC$. A fixed point $P$ lies on $AC$. The line passing through $P$ and parallel to $AO$ meets $BA$ at point $A_1$, the line passing through $P$ and parallel to $CO$ meets $BC$ at point $C_1$. Prove that the circumcenter of triangle $A_1BC_1$ moves along a straight line.

Ukrainian TYM Qualifying - geometry, 2013.6

Tags: geometry , Locus , Ukrainian TYM

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 $\omega$.

2018 Oral Moscow Geometry Olympiad, 5

Tags: geometry , Locus , fixed , midpoints , orthocenter

The circle circumscribed about an acute triangle $ABC$ and the vertex $C$ are fixed. Orthocenter $H$ moves in a circle with center at point $C$. Find the locus of the midpoints of the segments connecting the feet of altitudes drawn from vertices $A$ and $B$.

1973 IMO Shortlist, 1

Tags: 3D geometry , tetrahedron , sphere , Locus , Locus problems , IMO Shortlist , 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.

2024 ITAMO, 2

Tags: geometry , geometry proposed , Locus

We are given a unit square in the plane. A point $M$ in the plane is called [i]median [/i]if there exists points $P$ and $Q$ on the boundary of the square such that $PQ$ has length one and $M$ is the midpoint of $PQ$. Determine the geometric locus of all median points.

1965 IMO Shortlist, 5

Tags: geometry , parallelogram , Locus , Locus problems , IMO , IMO 1965

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 $\triangle OAB$.

1952 Moscow Mathematical Olympiad, 224-

Tags: Locus , geometry , Segment

You are given a segment $AB$. Find the locus of the vertices $C$ of acute-angled triangles $ABC$.

Ukrainian From Tasks to Tasks - geometry, 2012.2

Tags: geometry , Locus , Equilateral , isosceles

The triangle $ABC$ is equilateral. Find the locus of the points $M$ such that the triangles $ABM$ and $ACM$ are both isosceles.

2006 Sharygin Geometry Olympiad, 10.3

Tags: geometry , Locus , tangent circles , Tangents

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.

2012 Sharygin Geometry Olympiad, 4

Tags: Locus , right triangle , square , geometry

Consider a square. Find the locus of midpoints of the hypothenuses of rightangled triangles with the vertices lying on three different sides of the square and not coinciding with its vertices. (B.Frenkin)

1986 Tournament Of Towns, (123) 5

Tags: geometry , Locus , orthocenter , circle

Find the locus of the orthocentres (i.e. the point where three altitudes meet) of the triangles inscribed in a given circle . (A. Andjans, Riga)

1960 Putnam, A4

Tags: Putnam , geometry , Locus

Given two points, $P$ and $Q$, on the same side of a line $L$, the problem is to find a third point $R$ so that $PR+ RQ+RS$ is minimal, where $S$ is the unique point on $L$ such that $RS$ is perpendicular to $L.$ Consider all cases.

Ukraine Correspondence MO - geometry, 2006.3

Tags: geometry , Locus , orthocenter , Ukraine Correspondence

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

2019 Oral Moscow Geometry Olympiad, 5

Tags: geometry , perpendicular bisector , Locus , Locus problems , collinear

Given the segment $ PQ$ and a circle . A chord $AB$ moves around the circle, equal to $PQ$. Let $T$ be the intersection point of the perpendicular bisectors of the segments $AP$ and $BQ$. Prove that all points of $T$ thus obtained lie on one line.

1957 Moscow Mathematical Olympiad, 359

Tags: geometry , Locus , perpendicular

Straight lines $OA$ and $OB$ are perpendicular. Find the locus of endpoints $M$ of all broken lines $OM$ of length $\ell$ which intersect each line parallel to $OA$ or $OB$ at not more than one point.

2019 Gulf Math Olympiad, 1

Tags: geometry , trapezoid , Locus , area of a triangle , angle bisector

Let $ABCD$ be a trapezium (trapezoid) with $AD$ parallel to $BC$ and $J$ be the intersection of the diagonals $AC$ and $BD$. Point $P$ a chosen on the side $BC$ such that the distance from $C$ to the line $AP$ is equal to the distance from $B$ to the line $DP$. [i]The following three questions 1, 2 and 3 are independent, so that a condition in one question does not apply in another question.[/i] 1.Suppose that $Area( \vartriangle AJB) =6$ and that $Area(\vartriangle BJC) = 9$. Determine $Area(\vartriangle APD)$. 2. Find all points $Q$ on the plane of the trapezium such that $Area(\vartriangle AQB) = Area(\vartriangle DQC)$. 3. Prove that $PJ$ is the angle bisector of $\angle APD$.

2016 Sharygin Geometry Olympiad, 3

Tags: geometry , Locus , trapezoid , perpendicular

A trapezoid $ABCD$ and a line $\ell$ perpendicular to its bases $AD$ and $BC$ are given. A point $X$ moves along $\ell$. The perpendiculars from $A$ to $BX$ and from $D$ to $CX$ meet at point $Y$ . Find the locus of $Y$ . by D.Prokopenko

2004 All-Russian Olympiad Regional Round, 10.7

Tags: geometry , collinear , Locus , circles

Circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. At point $A$ to $\omega_1$ and $\omega_2$ the tangents $\ell_1$ and $\ell_2$ are drawn respectively. The points $T_1$ and $T_2$ are chosen respectively on the circles $\omega_1$ and $\omega_2$ so that the angular measures of the arcs $T_1A$ and $AT_2$ are equal (the measure of the circular arc is calculated clockwise). The tangent $t_1$ at the point $ T_1$ to the circle $\omega_1$ intersects $\ell_2$ at the point $M_1$. Similarly, the tangent $t_2$ at the point $T_2$ to the circle $\omega_2$ intersects $\ell_1$ at point $M_2$. Prove that the midpoints of the segments $M_1M_2$ are on the same a straight line that does not depend on the position of points $T_1$, $T_2$.

1970 Spain Mathematical Olympiad, 6

Tags: geometry , Locus , Circumcenter

Given a circle $\gamma$ and two points $A$ and $B$ in its plane. By $B$ passes a variable secant that intersects $\gamma$ at two points $M$ and $N$. Determine the locus of the centers of the circles circumscribed to the triangle $AMN$.

1963 Czech and Slovak Olympiad III A, 3

Tags: geometry , Locus , tangent circles , common tangent

A line $MN$ is given in the plane. Consider circles $k_1$, $k_2$ tangent to the line at points $M$, $N$, respectively, while touching each other externally. Let $X$ be the midpoint of the segment $PQ$, where $P$, $Q$ are in this order tangent points of the second common external tangent of the circles $k_1$, $k_2$. Find the locus of the points $X$ for all pairs of circles of the specified properties.

1969 IMO Longlists, 12

Tags: geometry , 3D geometry , tetrahedron , Locus , Locus problems , IMO Shortlist , IMO Longlist

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