Olympiad problems

1997 Croatia National Olympiad, Problem 2

Consider a circle $k$ and a point $K$ in the plane. For any two distinct points $P$ and $Q$ on $k$, denote by $k'$ the circle through $P,Q$ and $K$. The tangent to $k'$ at $K$ meets the line $PQ$ at point $M$. Describe the locus of the points $M$ when $P$ and $Q$ assume all possible positions.

1953 Moscow Mathematical Olympiad, 239

Tags: trigonometry , analytic geometry , locus

On the plane find the locus of points whose coordinates satisfy $sin(x + y) = 0$.

V Soros Olympiad 1998 - 99 (Russia), 11.4

Tags: geometry , locus , tangent

A chord $AB$ is drawn in a circle. On its extensions beyond points $A$ and $B$, points $P$ and $Q$ respectively are taken such that $AP = BQ$. Through $P$ and $Q$ two tangents to the circle are drawn, intersecting at point $M$. Find the locus of points $M$ ($P$ and $Q$ move along a straight line and for any $P$ and $Q$ all possible pairs of tangents are taken, which determine four points from the desired locus of points) .

1973 IMO Shortlist, 5

Tags: incircle , locus , 3d geometry , locus problems , geometry

A circle of radius 1 is located in a right-angled trihedron and touches all its faces. Find the locus of centers of such circles.

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.

2014 Hanoi Open Mathematics Competitions, 7

Tags: locus , tangent circle , geometry

Let two circles $C_1,C_2$ with different radius be externally tangent at a point $T$. Let $A$ be on $C_1$ and $B$ be on $C_2$, with $A,B \ne T$ such that $\angle ATB = 90^o$. (a) Prove that all such lines $AB$ are concurrent. (b) Find the locus of the midpoints of all such segments $AB$.

Estonia Open Senior - geometry, 2019.2.5

Tags: locus , equilateral , geometry

The plane has a circle $\omega$ and a point $A$ outside it. For any point $C$, the point $B$ on the circle $\omega$ is defined such that $ABC$ is an equilateral triangle with vertices $A, B$ and $C$ listed clockwise. Prove that if point $B$ moves along the circle $\omega$, then point $C$ also moves along a circle.

1986 Traian Lălescu, 2.2

Tags: analytic geometry , geometry , locus , locus problems

Let be a line $ d: 3x+4y-5=0 $ on a Cartesian plane. We mark with $ \mathcal{L} $ de locus of the planar points $ P $ such that the distance from $ P $ to $ d $ is double the distance from $ P $ to the origin. Let be $ B_{\lambda } ,C_{\lambda }\in\mathcal{L} $ such that $ C_{\lambda } -B_{\lambda } +\lambda =0. $ Find the locus of the middlepoints of the segments $ B_{\lambda }C_{\lambda }, $ if $ \lambda\in\mathbb{R} $ is variable.

1969 IMO Longlists, 1

Tags: conic , parabola , locus , locus problems , geometry

$(BEL 1)$ A parabola $P_1$ with equation $x^2 - 2py = 0$ and parabola $P_2$ with equation $x^2 + 2py = 0, p > 0$, are given. A line $t$ is tangent to $P_2.$ Find the locus of pole $M$ of the line $t$ with respect to $P_1.$

1942 Putnam, A1

Tags: locus , square

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.

1978 Romania Team Selection Test, 5

Tags: pure geometry , locus , geometry

Find locus of points $ M $ inside an equilateral triangle $ ABC $ such that $$ \angle MBC+\angle MCA +\angle MAB={\pi}/{2}. $$

1986 IMO Shortlist, 1

Tags: rotation , polygon , locus , locus problems , geometry

Let $A,B$ be adjacent vertices of a regular $n$-gon ($n\ge5$) with center $O$. A triangle $XYZ$, which is congruent to and initially coincides with $OAB$, moves in the plane in such a way that $Y$ and $Z$ each trace out the whole boundary of the polygon, with $X$ remaining inside the polygon. Find the locus of $X$.

1978 IMO, 2

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

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.

2002 Junior Balkan Team Selection Tests - Moldova, 7

Tags: angle , geometry , locus , perimeter

The side of the square $ABCD$ has a length equal to $1$. On the sides $(BC)$ ¸and $(CD)$ take respectively the arbitrary points $M$ and $N$ so that the perimeter of the triangle $MCN$ is equal to $2$. a) Determine the measure of the angle $\angle MAN$. b) If the point $P$ is the foot of the perpendicular taken from point $A$ to the line $MN$, determine the locus of the points $P$.

III Soros Olympiad 1996 - 97 (Russia), 10.7

Tags: locus , geometry

Let $A$ be a fixed point on a circle, $B$ and$ C$ be arbitrary points on the circle different from $A$ and at different distances. The bisector of the angle $\angle BAC$ intersects the chord $BC$ and the circle at points $K$ and $P$, $D$ is the projection of $A$ onto the straight line $BC$. A circle passing through points $K$, $P$ and $D$ intersects the straight line $AD$ for the second time at point $M$. Find the locus of points $M$.

1949-56 Chisinau City MO, 32

Tags: right triangle , midpoint , locus , geometry

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.

2017 Tuymaada Olympiad, 8

Tags: analytic geometry , locus , coordinate geometry , geometry

Two points $A$ and $B$ are given in the plane. A point $X$ is called their [i]preposterous midpoint[/i] if there is a Cartesian coordinate system in the plane such that the coordinates of $A$ and $B$ in this system are non-negative, the abscissa of $X$ is the geometric mean of the abscissae of $A$ and $B$, and the ordinate of $X$ is the geometric mean of the ordinates of $A$ and $B$. Find the locus of all the [i]preposterous midpoints[/i] of $A$ and $B$. (K. Tyschu)

1980 Spain Mathematical Olympiad, 7

Tags: analytic geometry , locus , geometry , area

The point $M$ varies on the segment $AB$ that measures $2$ m. a) Find the equation and the graphical representation of the locus of the points of the plane whose coordinates, $x$, and $y$, are, respectively, the areas of the squares of sides $AM$ and $MB$ . b) Find out what kind of curve it is. (Suggestion: make a $45^o$ axis rotation). c) Find the area of the enclosure between the curve obtained and the coordinate axes.

1995 Tournament Of Towns, (477) 1

Tags: parallelogram , geometry , angle bisector , locus

If P is a point inside a convex quadrilateral $ABCD$, let the angle bisectors of $\angle APB$, $\angle BPC$, $\angle CPD$ and $\angle DPA$ meet $AB$, $BC$, $CD$ and $DA$ at $K$, $L$, $M$ and $N$ respectively. (a) Find a point $P$ such that $KLMN$ is a parallelogram. (b) Find the locus of all such points $P$. (S Tokarev)

2004 Oral Moscow Geometry Olympiad, 3

Tags: geometry , locus , equal angles

Given a square $ABCD$. Find the locus of points $M$ such that $\angle AMB = \angle CMD$.

1997 Argentina National Olympiad, 5

Tags: equal area , locus , area of a triangle , geometry

Given two non-parallel segments $AB$ and $CD$ on the plane, find the locus of points $P$ on the plane such that the area of triangle $ABP$ is equal to the area of triangle $CDP$.

2007 Sharygin Geometry Olympiad, 18

Tags: orthocenter , locus , locus problems , circumcenter , circumcircle , geometry

Determine the locus of vertices of triangles which have prescribed orthocenter and center of circumcircle.

1972 Czech and Slovak Olympiad III A, 2

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

Let $ABCDA'B'C'D'$ be a cube (where $ABCD$ is a square and $AA'\parallel BB'\parallel CC'\parallel DD'$). Furthermore, let $\mathcal R$ be a rotation (with respect some line) that maps vertex $A$ to $B.$ Find the set of all images $X=\mathcal R(C)$ such that $X$ lies on the surface of the cube for some rotation $\mathcal R(A)=B.$

Estonia Open Junior - geometry, 1999.1.2

Tags: geometry , locus , isosceles triangle , isosceles

Two different points $X$ and $Y$ are chosen in the plane. Find all the points $Z$ in this plane for which the triangle $XYZ$ is isosceles.

1966 IMO Longlists, 28

Tags: locus , locus problems , geometry , circles

In the plane, consider a circle with center $S$ and radius $1.$ Let $ABC$ be an arbitrary triangle having this circle as its incircle, and assume that $SA\leq SB\leq SC.$ Find the locus of [b]a.)[/b] all vertices $A$ of such triangles; [b]b.)[/b] all vertices $B$ of such triangles; [b]c.)[/b] all vertices $C$ of such triangles.