Olympiad problems

2016 Sharygin Geometry Olympiad, P10

Point $X$ moves along side $AB$ of triangle $ABC$, and point $Y$ moves along its circumcircle in such a way that line $XY$ passes through the midpoint of arc $AB$. Find the locus of the circumcenters of triangles $IXY$ , where I is the incenter of $ ABC$.

2012 Ukraine Team Selection Test, 4

Tags: geometry , locus , incircle , isosceles

Given an isosceles triangle $ABC$ ($AB = AC$), the inscribed circle $\omega$ touches its sides $AB$ and $AC$ at points $K$ and $L$, respectively. On the extension of the side of the base $BC$, towards $B$, an arbitrary point $M$. is chosen. Line $M$ intersects $\omega$ at the point $N$ for the second time, line $BN$ intersects the second point $\omega$ at the point $P$. On the line $PK$, there is a point $X$ such that $K$ lies between $P$ and $X$ and $KX = KM$. Determine the locus of the point $X$.

1998 Belarus Team Selection Test, 1

Tags: geometry , angle , locus , circles

Two circles $S_1$ and $S_2$ intersect at different points $P,Q$. The arc of $S_1$ lying inside $S_2$ measures $2a$ and the arc of $S_2$ lying inside $S_1$ measures $2b$. Let $T$ be any point on $S_1$. Let $R,S$ be another points of intersection of $S_2$ with $TP$ and $TQ$ respectively. Let $a+2b<\pi$ . Find the locus of the intersection points of $PS$ and $RQ$. S.Shikh

1965 Vietnam National Olympiad, 2

Tags: tetrahedron , geometry , locus , 3-dimensional geometry , 3d geometry

$AB$ and $CD$ are two fixed parallel chords of the circle $S$. $M$ is a variable point on the circle. $Q$ is the intersection of the lines $MD$ and $AB$. $X$ is the circumcenter of the triangle $MCQ$. Find the locus of $X$. What happens to $X$ as $M$ tends to (1) $D$, (2) $C$? Find a point $E$ outside the plane of $S$ such that the circumcenter of the tetrahedron $MCQE$ has the same locus as $X$.

2022 Oral Moscow Geometry Olympiad, 5

Tags: geometry , locus

Given a circle and a straight line $AB$ passing through its center (points $A$ and $B$ are fixed, $A$ is outside the circle, and $B$ is inside). Find the locus of the intersection of lines $AX$ and $BY$, where $XY$ is an arbitrary diameter of the circle. (A. Akopyan, A. Zaslavsky)

1987 Tournament Of Towns, (151) 2

Tags: geometry , rhombus , angle , locus

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

1966 IMO Shortlist, 55

Tags: geometry , centroid , locus , locus problems

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 $[40^\circ, 70^\circ].$

1941 Moscow Mathematical Olympiad, 089

Tags: geometry , skew , locus

Given two skew perpendicular lines in space, find the set of the midpoints of all segments of given length with the endpoints on these lines.

2006 Hanoi Open Mathematics Competitions, 6

Tags: locus , hexagon , geometry

The figure $ABCDEF$ is a regular hexagon. Find all points $M$ belonging to the hexagon such that Area of triangle $MAC =$ Area of triangle $MCD$.

1956 Moscow Mathematical Olympiad, 325

Tags: midpoint , locus , geometry , equal segments

On sides $AB$ and $CB$ of $\vartriangle ABC$ there are drawn equal segments, $AD$ and $CE$, respectively, of arbitrary length (but shorter than min($AB,BC$)). Find the locus of midpoints of all possible segments $DE$.

2000 Tournament Of Towns, 3

Tags: geometry , locus , fixed , rectangle , circles

$A$ is a fixed point inside a given circle. Determine the locus of points $C$ such that $ABCD$ is a rectangle with $B$ and $D$ on the circumference of the given circle. (M Panov)

1995 Grosman Memorial Mathematical Olympiad, 4

Tags: geometry , circles , locus , intersecting circles

Two given circles $\alpha$ and $\beta$ intersect each other at two points. Find the locus of the centers of all circles that are orthogonal to both $\alpha$ and $\beta$.

1982 Bulgaria National Olympiad, Problem 6

Tags: geometry , triangle , square , centroid , triangle centers , locus

Find the locus of centroids of equilateral triangles whose vertices lie on sides of a given square $ABCD$.

1999 Estonia National Olympiad, 5

Tags: geometry , locus , midpoint , equilateral

Let $C$ be an interior point of line segment $AB$. Equilateral triangles $ADC$ and $CEB$ are constructed to the same side from $AB$. Find all points which can be the midpoint of the segment $DE$.

1984 Spain Mathematical Olympiad, 6

Tags: geometry , analytic geometry , locus

Consider the circle $\gamma$ with center at point $(0,3)$ and radius $3$, and a line $r$ parallel to the axis $Ox$ at a distance $3$ from the origin. A variable line through the origin meets $\gamma$ at point $M$ and $r$ at point $P$. Find the locus of the intersection point of the lines through $M$ and $P$ parallel to $Ox$ and $Oy$ respectively.

1966 IMO Longlists, 55

Tags: geometry , centroid , locus , locus problems

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 $[40^\circ, 70^\circ].$

1978 Romania Team Selection Test, 2

Tags: rotation , tetrahedron , geometric transformation , 3d geometry , locus , pure geometry , geometry

Points $ A’,B,C’ $ are arbitrarily taken on edges $ SA,SB, $ respectively, $ SC $ of a tetrahedron $ SABC. $ Plane forrmed by $ ABC $ intersects the plane $ \rho , $ formed by $ A’B’C’, $ in a line $ d. $ Prove that, meanwhile the plane $ \rho $ rotates around $ d, $ the lines $ AA’,BB’ $ and $ CC’ $ are, and remain concurrent. Find de locus of the respective intersections.

1964 Vietnam National Olympiad, 3

Tags: 3-dimensional geometry , locus , geometry

Let $P$ be a plane and two points $A \in (P),O \notin (P)$. For each line in $(P)$ through $A$, let $H$ be the foot of the perpendicular from $O$ to the line. Find the locus $(c)$ of $H$. Denote by $(C)$ the oblique cone with peak $O$ and base $(c)$. Prove that all planes, either parallel to $(P)$ or perpendicular to $OA$, intersect $(C)$ by circles. Consider the two symmetric faces of $(C)$ that intersect $(C)$ by the angles $\alpha$ and $\beta$ respectively. Find a relation between $\alpha$ and $\beta$.

Swiss NMO - geometry, 2008.5

Tags: locus , square , geometry

Let $ABCD$ be a square with side length $1$. Find the locus of all points $P$ with the property $AP\cdot CP + BP\cdot DP = 1$.

2014 Contests, 1

Tags: trigonometry , orthocenter , locus , geometry

Let $k$ be a given circle and $A$ is a fixed point outside $k$. $BC$ is a diameter of $k$. Find the locus of the orthocentre of $\triangle ABC$ when $BC$ varies. [i]Proposed by T. Vitanov, E. Kolev[/i]

1970 Vietnam National Olympiad, 4

Tags: construction , locus , tangent , diameter , geometry

$AB$ 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 $Q"$.

1961 All Russian Mathematical Olympiad, 006

Tags: geometry , equilateral , locus

a) Points $A$ and $B$ move uniformly and with equal angle speed along the circumferences with $O_a$ and $O_b$ centres (both clockwise). Prove that a vertex $C$ of the equilateral triangle $ABC$ also moves along a certain circumference uniformly. b) The distance from the point $P$ to the vertices of the equilateral triangle $ABC$ equal $|AP|=2, |BP|=3$. Find the maximal value of $CP$.

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 $PXY$.

Kvant 2024, M2784

Tags: geometry , locus

The bisectors $AD{}$ and $BE{}$ were drawn in the triangle $ABC{}$ and they intersected at point $I{}.$ Then everything was erased, leaving only the points $D{}$ and $E{}.$ Find the set of possible positions of the point $I{}.$ [i]Proposed by M. Didin[/i]

1965 German National Olympiad, 4

Tags: geometry , inradius , locus , distance , regular polygon

Find the locus of points in the plane, the sum of whose distances from the sides of a regular polygon is five times the inradius of the pentagon.