Olympiad problems

2007 Sharygin Geometry Olympiad, 3

Tags: geometry , Circumcenter , Locus , Locus problems , circumcircle , circles

Given two circles intersecting at points $P$ and $Q$. Let C be an arbitrary point distinct from $P$ and $Q$ on the former circle. Let lines $CP$ and $CQ$ intersect again the latter circle at points A and B, respectively. Determine the locus of the circumcenters of triangles $ABC$.

1986 Bulgaria National Olympiad, Problem 5

Tags: Locus , geometry , circles

Let $A$ be a fixed point on a circle $k$. Let $B$ be any point on $k$ and $M$ be a point such that $AM:AB=m$ and $\angle BAM=\alpha$, where $m$ and $\alpha$ are given. Find the locus of point $M$ when $B$ describes the circle $k$.

1996 Rioplatense Mathematical Olympiad, Level 3, 4

Tags: geometry , circle , Locus

Let $S$ be the circle of center $O$ and radius $R$, and let $A, A'$ be two diametrically opposite points in $S$. Let $P$ be the midpoint of $OA'$ and $\ell$ a line passing through $P$, different from $AA '$ and from the perpendicular on $AA '$. Let $B$ and $C$ be the intersection points of $\ell$ with $S$ and let $M$ be the midpoint of $BC$. a) Let $H$ be the foot of the altitude from $A$ in the triangle $ABC$. Let $D$ be the intersection point of the line $A'M$ with $AH$. Determine the locus of point $D$ while $\ell$ varies . b) Line $AM$ intersects $OD$ at $I$. Prove that $2 OI = ID$ and determine the locus of point $I$ while $\ell$ varies .

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]

1987 Mexico National Olympiad, 3

Tags: geometry , Locus , projections

Consider two lines $\ell$ and $\ell ' $ and a fixed point $P$ equidistant from these lines. What is the locus of projections $M$ of $P$ on $AB$, where $A$ is on $\ell $, $B$ on $\ell ' $, and angle $\angle APB$ is right?

2015 Sharygin Geometry Olympiad, P20

Tags: conics , ellipse , Locus , geometry , circumcircle

Given are a circle and an ellipse lying inside it with focus $C$. Find the locus of the circumcenters of triangles $ABC$, where $AB$ is a chord of the circle touching the ellipse.

2002 Moldova Team Selection Test, 3

Tags: geometry , Locus , minimum

A triangle $ABC$ is inscribed in a circle $G$. For any point $M$ inside the triangle, $A_1$ denotes the intersection of the ray $AM$ with $G$. Find the locus of point $M$ for which $\frac{BM\cdot CM}{MA_1}$ is minimal, and find this minimum value.

1966 IMO Shortlist, 17

Tags: geometry , parallelogram , vector , Locus problems , Locus , IMO Longlist , IMO Shortlist

Let $ABCD$ and $A^{\prime }B^{\prime}C^{\prime }D^{\prime }$ be two arbitrary parallelograms in the space, and let $M,$ $N,$ $P,$ $Q$ be points dividing the segments $AA^{\prime },$ $BB^{\prime },$ $CC^{\prime },$ $DD^{\prime }$ in equal ratios. [b]a.)[/b] Prove that the quadrilateral $MNPQ$ is a parallelogram. [b]b.)[/b] What is the locus of the center of the parallelogram $MNPQ,$ when the point $M$ moves on the segment $AA^{\prime }$ ? (Consecutive vertices of the parallelograms are labelled in alphabetical order.

1949-56 Chisinau City MO, 46

Tags: Locus , ratio , geometry

Determine the locus of points, for whom the ratio of the distances to two given points has a constant value.

Estonia Open Senior - geometry, 1995.2.4

Tags: Locus , geometry , Sum , distance

Find all points on the plane such that the sum of the distances of each of the four lines defined by the unit square of that plane is $4$.

1998 Mexico National Olympiad, 2

Tags: geometry , Locus , circles

Rays $l$ and $m$ forming an angle of $a$ are drawn from the same point. Let $P$ be a fixed point on $l$. For each circle $C$ tangent to $l$ at $P$ and intersecting $m$ at $Q$ and $R$, let $T$ be the intersection point of the bisector of angle $QPR$ with $C$. Describe the locus of $T$ and justify your answer.

2014 Bulgaria National Olympiad, 1

Tags: trigonometry , geometry , orthocenter , Locus

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]

2008 Grigore Moisil Intercounty, 2

Tags: Locus , geometry , quadrilateral

Given a convex quadrilateral $ ABCD, $ find the locus of points $ X $ that verify the qualities: $$ XA^2+XB^2+CD^2=XB^2+XC^2+DA^2=XC^2+XD^2+AB^2=XD^2+XA^2+BC^2 $$ [i]Maria Pop[/i]

2010 Cuba MO, 8

Tags: geometry , parallelogram , Locus

Let $ABCDE$ be a convex pentagon that has $AB < BC$, $AE <ED$ and $AB + CD + EA = BC + DE$. Variable points $F,G$ and $H$ are taken that move on the segments $BC$, $CD$ and $OF$ respectively . $B'$ is defined as the projection of $B$ on $AF$, $C'$ as the projection of $C$ on $FG$, $D'$ as the projection of $D$ on $GH$ and $E'$ as the projection of $E$ onto $HA$. Prove that there is at least one quadrilateral $B'C'D'E'$ when $F,G$ and $H$ move on their sides, which is a parallelogram.

2007 Sharygin Geometry Olympiad, 4

Tags: geometry , Locus , orthocenter , Locus problems

Determine the locus of orthocenters of triangles, given the midpoint of a side and the feet of the altitudes drawn on two other sides.

III Soros Olympiad 1996 - 97 (Russia), 10.7

Tags: geometry , Locus

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 $M$.

2020 Tournament Of Towns, 5

Tags: geometry , circles , Locus

Given are two circles which intersect at points $P$ and $Q$. Consider an arbitrary line $\ell$ through $Q$, let the second points of intersection of this line with the circles be $A$ and $B$ respectively. Let $C$ be the point of intersection of the tangents to the circles in those points. Let $D$ be the intersection of the line $AB$ and the bisector of the angle $CPQ$. Prove that all possible $D$ for any choice of $\ell$ lie on a single circle. Alexey Zaslavsky

2021 Stars of Mathematics, 3

Tags: geometry , Locus , romania

Let $ABC$ be a triangle, let its $A$-symmedian cross the circle $ABC$ again at $D$, and let $Q$ and $R$ be the feet of the perpendiculars from $D$ on the lines $AC$ and $AB$, respectively. Consider a variable point $X$ on the line $QR$, different from both $Q$ and $R$. The line through $X$ and perpendicular to $DX$ crosses the lines $AC$ and $AB$ at $V$ and $W$, respectively. Determine the geometric locus of the midpoint of the segment $VW$. [i]Adapted from American Mathematical Monthly[/i]

2017 Sharygin Geometry Olympiad, P20

Tags: geometry , Locus , geometric transformation , reflection

Given a right-angled triangle $ABC$ and two perpendicular lines $x$ and $y$ passing through the vertex $A$ of its right angle. For an arbitrary point $X$ on $x$ define $y_B$ and $y_C$ as the reflections of $y$ about $XB$ and $ XC $ respectively. Let $Y$ be the common point of $y_b$ and $y_c$. Find the locus of $Y$ (when $y_b$ and $y_c$ do not coincide).

1997 Argentina National Olympiad, 5

Tags: geometry , Locus , area of a triangle , equal areas

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 $CDP$.

Cono Sur Shortlist - geometry, 2003.G6

Tags: geometry , Locus , Parallel Lines

Let $L_1$ and $L_2$ be two parallel lines and $L_3$ a line perpendicular to $L_1$ and $L_2$ at $H$ and $P$, respectively. Points $Q$ and $R$ lie on $L_1$ such that $QR = PR$ ($Q \ne H$). Let $d$ be the diameter of the circle inscribed in the triangle $PQR$. Point $T$ lies $L_2$ in the same semiplane as $Q$ with respect to line $L_3$ such that $\frac{1}{TH}= \frac{1}{d}- \frac{1}{PH}$ . Let $X$ be the intersection point of $PQ$ and $TH$. Find the locus of the points $X$ as $Q$ varies on $L_1$.

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 $\mathcal R(A)=B.$

2022 Oral Moscow Geometry Olympiad, 5

Tags: Locus , geometry

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)

2002 Junior Balkan Team Selection Tests - Moldova, 7

Tags: geometry , Locus , perimeter , angle

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 $P$.

2016 Tournament Of Towns, 2

Tags: geometry , Locus , circles

On plane there is fixed ray $s$ with vertex $A$ and a point $P$ not on the line which contains $s$. We choose a random point $K$ which lies on ray. Let $N$ be a point on a ray outside $AK$ such that $NK=1$. Let $M$ be a point such that $NM=1,M \in PK$ and $M!=K.$ Prove that all lines $NM$, provided by some point $K$, touch some fixed circle.