Olympiad problems

1969 Spain Mathematical Olympiad, 2

Find the locus of the affix $M$, of the complex number $z$, so that it is aligned with the affixes of $i$ and $iz$ .

1986 IMO Shortlist, 16

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 $X$.

1959 Polish MO Finals, 5

Tags: locus , geometry

In the plane of the triangle $ ABC $ a straight line moves which intersects the sides $ AC $ and $ BC $ in such points $ D $ and $ E $ that $ AD = BE $. Find the locus of the midpoint $ M $ of the segment $ DE $.

1968 Czech and Slovak Olympiad III A, 3

Tags: geometry , geometric transformation , reflection , locus

Two segment $AB,CD$ of the same length are given in plane such that lines $AB,CD$ are not parallel. Consider a point $S$ with the following property: the image of segment $AB$ under point reflection with respect to $S$ is identical to the mirror-image of segment $CD$ with respect to some axis. Find the locus of all such points $S.$

IV Soros Olympiad 1997 - 98 (Russia), 9.5

Tags: geometry , rotation , locus , geometric transformation

Given triangle $ABC$. Find the locus of points $M$ such that there is a rotation with center at $M$ that takes $C$ to a certain point on side $AB$.

1940 Moscow Mathematical Olympiad, 061

Tags: locus , distance , geometry

Given two lines on a plane, find the locus of all points with the difference between the distance to one line and the distance to the other equal to the length of a given segment.

1995 Tournament Of Towns, (477) 1

Tags: angle bisector , locus , parallelogram , geometry

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)

1975 Czech and Slovak Olympiad III A, 5

Tags: triangle , geometry , locus , isosceles , square

Let a square $\mathbf P=P_1P_2P_3P_4$ be given in the plane. Determine the locus of all vertices $A$ of isosceles triangles $ABC,AB=BC$ such that the vertices $B,C$ are points of the square $\mathbf P.$

2023 Sharygin Geometry Olympiad, 19

Tags: geometry , locus , circle intersections

A cyclic quadrilateral $ABCD$ is given. An arbitrary circle passing through $C$ and $D$ meets $AC,BC$ at points $X,Y$ respectively. Find the locus of common points of circles $CAY$ and $CBX$.

1986 IMO, 1

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 $X$.

1982 Spain Mathematical Olympiad, 8

Tags: analytic geometry , geometry , locus

Given a set $C$ of points in the plane, it is called the distance of a point $P$ from the plane to the set $C$ at the smallest of the distances from $P$ to each of the points of $C$. Let the sets be $C = \{A,B\}$, with $A = (1, 0)$ and $B = (2, 0)$; and $C'= \{A',B'\}$ with $A' = (0, 1)$ and $B' = (0, 7)$, in an orthogonal reference system. Find and draw the set $M$ of points in the plane that are equidistant from $C$ and $C'$ . Study whether the function whose graph is the set $M$ previously obtained is derivable.

1949-56 Chisinau City MO, 61

Tags: geometry , locus , projection , 3d geometry

Find the locus of the projections of a given point on all planes containing another point $B$.

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 $DE$.

VII Soros Olympiad 2000 - 01, 10.5

Tags: geometry , incenter , locus , angle

An acute-angled triangle $ABC$ is given. Points $A_1, B_1$ and $C_1$ are taken on its sides $BC, CA$ and $AB$, respectively, such that $\angle B_1A_1C_1 + 2 \angle BAC = 180^o$, $\angle A_1C_1B_1 + 2 \angle ACB = 180^o$, $\angle C_1B_1A_1 + 2 \angle CBA = 180^o$. Find the locus of the centers of the circles inscribed in triangles $A_1B_1C_1$ (all kinds of such triangles are considered).

2007 Nicolae Coculescu, 4

Tags: locus , analytic geometry , infimum , geometry

Let $ M $ be a point in the interior of a triangle $ ABC, $ let $ D $ be the intersection of $ AM $ with $ BC, $ let $ E $ be the intersection of $ M $ with AC, let $ F $ be the intersection of $ CM $ with $ AB. $ Knowing that the expression $$ \frac{MA}{MD}\cdot \frac{MB}{ME}\cdot \frac{MC}{MF} $$ is minimized, describe the point $ M. $

1962 Czech and Slovak Olympiad III A, 3

Tags: geometry , 3d geometry , line , locus , circles

Let skew lines $PM, QN$ be given such that $PM\perp PQ\perp QN$. Let a plane $\sigma\perp PQ$ containing the midpoint $O$ of segment $PQ$ be given and in it a circle $k$ with center $O$ and given radius $r$. Consider all segments $XY$ with endpoint $X, Y$ on lines $PM, QN$, respectively, which contain a point of $k$. Show that segments $XY$ have the same length. Find the locus of all such points $X$.

1980 All Soviet Union Mathematical Olympiad, 287

Tags: geometry , locus , convex , midpoint , area

The points $M$ and $P$ are the midpoints of $[BC]$ and $[CD]$ sides of a convex quadrangle $ABCD$. It is known that $|AM| + |AP| = a$. Prove that $ABCD$ has area less than $\frac{a^2}{2}$.

1935 Moscow Mathematical Olympiad, 014

Tags: geometry , locus , locus in space , cube

Find the locus of points on the surface of a cube that serve as the vertex of the smallest angle that subtends the diagonal.

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 $d$.

1996 Romania National Olympiad, 4

Tags: geometry , locus

In the triangle $ABC$ the incircle $J$ touches the sides $BC$, $CA$, $AB$ in $D$, $E$, $F$, respectively. The segments $(BE)$ and $(CF)$ intersect $J$ in $G,H$. If $B$ and $C$ are fixed points, find the loci of points $A, D, E, F, G, H$ if $GH \parallel BC$ and the loci of the same points if $BCHG$ is an inscriptible quadrilateral.

2017 Tuymaada Olympiad, 8

Tags: locus , coordinate geometry , analytic 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)

1966 IMO Longlists, 28

Tags: geometry , locus , locus problems , 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.