Olympiad problems

1981 All Soviet Union Mathematical Olympiad, 326

The segments $[AD], [BE]$ and $[CF]$ are the side edges of the right triangle prism. (the equilateral triangle is a base) Find all the points in its base $ABC$, situated on the equal distances from the $(AE), (BF)$ and $(CD)$ lines.

1984 Spain Mathematical Olympiad, 6

Tags: locus , analytic geometry , geometry

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.

1985 Czech And Slovak Olympiad IIIA, 4

Tags: geometry , locus

Two straight lines $p, q$ are given in the plane and on the straight line $q$ there is a point $F$, $F \not\in p$. Determine the set of all points $X$ that can be obtained by this construction: In the plane we choose a point $S$ that lies neither on $p$ nor on $q$, and we construct a circle $k$ with center $S$ that is tangent to the line $p$. On the circle $k$ we choose a point $T$ such that so that $ST \parallel q$. If the line $FT$ intersects the line $p$ at the point $U$, $X$ is the intersection of the lines $SU$ and $q$

1972 Czech and Slovak Olympiad III A, 2

Tags: geometry , 3d geometry , geometric transformation , rotation , 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 $\mathcal R(A)=B.$

1960 Putnam, A4

Tags: locus , geometry

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.

1969 IMO Shortlist, 12

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

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

1960 IMO Shortlist, 5

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

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 $ZY=2XZ$.

Kyiv City MO 1984-93 - geometry, 1992.8.3

Tags: geometry , centroid , locus

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

1979 Romania Team Selection Tests, 1.

Tags: geometry , locus

Let $\triangle ABC$ be a triangle with $\angle BAC=60^\circ$, $M$ be a point in its interior and $A',\, B',\, C'$ be the orthogonal projections of $M$ on the sides $BC,\, CA,\, AB$. Determine the locus of $M$ when the sum $A'B+B'C+C'A$ is constant. [i]Horea Călin Pop[/i]

1969 IMO Shortlist, 1

Tags: parabola , locus , locus problems , conic , 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 $P_1.$

2016 Sharygin Geometry Olympiad, P10

Tags: geometry , circumcircle , incenter , locus

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$.

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.

1973 IMO Shortlist, 1

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

Kyiv City MO Seniors 2003+ geometry, 2007.10.3

Tags: geometry , circumcircle , incircle , angle bisector , locus

The points $ P, Q$ are given on the plane, which are the points of intersection of the angle bisector $AL$ of some triangle $ABC$ with an inscribed circle, and the point $W$ is the intersection of the angle bisector $AL$ with a circumscribed circle other than the vertex $A$. a) Find the geometric locus of the possible location of the vertex $A$ of the triangle $ABC$. b) Find the geometric locus of the possible location of the vertex $B$ of the triangle $ABC$.

1960 IMO, 7

Tags: geometry , trapezoid , locus , locus problems

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.

2006 Hanoi Open Mathematics Competitions, 6

Tags: geometry , hexagon , locus

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

1965 IMO Shortlist, 5

Tags: geometry , parallelogram , locus , locus problems

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 $\triangle OAB$.

V Soros Olympiad 1998 - 99 (Russia), 9.5

Tags: geometry , locus , isosceles

An angle with vertex $A$ is given on the plane. Points $K$ and $P$ are selected on its sides so that $AK + AP = a$, where $a$ is a given segment. Let $M$ be a point on the plane such that the triangle $KPM$ is isosceles with the base $KP$ and the angle at the vertex $M$ equal to the given angle. Find the locus of points $M$ (as $K$ and $P$ move).

2004 Oral Moscow Geometry Olympiad, 3

Tags: equal angles , geometry , locus

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

1946 Moscow Mathematical Olympiad, 122

Tags: sum , locus , ratio , geometry , area

On the sides $PQ, QR, RP$ of $\vartriangle PQR$ segments $AB, CD, EF$ are drawn. Given a point $S_0$ inside triangle $\vartriangle PQR$, find the locus of points $S$ for which the sum of the areas of triangles $\vartriangle SAB$, $\vartriangle SCD$ and $\vartriangle SEF$ is equal to the sum of the areas of triangles $\vartriangle S_0AB$, $\vartriangle S_0CD$, $\vartriangle S0EF$. Consider separately the case $$\frac{AB}{PQ }= \frac{CD}{QR} = \frac{EF}{RP}.$$

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.

1986 IMO Longlists, 33

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

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 $X$.

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.

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 $d$.

1978 IMO Shortlist, 13

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

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.