Olympiad problems

2009 Federal Competition For Advanced Students, P2, 3

Let $P$ be the point in the interior of $\vartriangle ABC$. Let $D$ be the intersection of the lines $AP$ and $BC$ and let $A'$ be the point such that $\overrightarrow{AD} = \overrightarrow{DA'}$. The points $B'$ and $C'$ are defined in the similar way. Determine all points $P$ for which the triangles $A'BC, AB'C$, and $ABC'$ are congruent to $\vartriangle ABC$.

2007 Hanoi Open Mathematics Competitions, 13

Tags: locus , area , geometry

Let be given triangle $ABC$. Find all points $M$ such that area of $\vartriangle MAB$= area of $\vartriangle MAC$

1956 Moscow Mathematical Olympiad, 339

Tags: geometry , locus , projection

Find the union of all projections of a given line segment $AB$ to all lines passing through a given point $O$.

Ukrainian TYM Qualifying - geometry, 2015.23

Tags: geometry , locus

An acute-angled triangle $ABC$ is given, through the vertices $B$ and $C$ of which a circle $\Omega$, $A \notin \Omega$, is drawn. We consider all points $P \in \Omega$, that do not lie on none of the lines $AB$ and $AC$ and for which the common tangents of the circumscribed circles of triangles $APB$ and $APC$ are not parallel. Let $X_P$ be the point of intersection of such two common tangents. a) Prove that the locus of points $X_P$ lies to some two lines. b) Prove that if the circle $\Omega$ passes through the orthocenter of the triangle $ABC$, then one of these lines is the line $BC$.

1953 Polish MO Finals, 2

Tags: geometry , locus , rectangle , inscribed

Find the geometric locus of the center of a rectangle whose vertices lie on the perimeter of a given triangle.

1955 Moscow Mathematical Olympiad, 293

Tags: locus , equal segments , geometry

Consider a quadrilateral $ABCD$ and points $K, L, M, N$ on sides $AB, BC, CD$ and $AD$, respectively, such that $KB = BL = a, MD = DN = b$ and $KL \nparallel MN$. Find the set of all the intersection points of $KL$ with $MN$ as $a$ and $b$ vary.

2019 Oral Moscow Geometry Olympiad, 5

Tags: geometry , perpendicular bisector , locus , locus problems , collinear

Given the segment $ PQ$ and a circle . A chord $AB$ moves around the circle, equal to $PQ$. Let $T$ be the intersection point of the perpendicular bisectors of the segments $AP$ and $BQ$. Prove that all points of $T$ thus obtained lie on one line.

1971 Czech and Slovak Olympiad III A, 5

Tags: geometry , equilateral , locus , triangle

Let $ABC$ be a given triangle. Find the locus $\mathbf M$ of all vertices $Z$ such that triangle $XYZ$ is equilateral where $X$ is any point of segment $AB$ and $Y\neq X$ lies on ray $AC.$

1949-56 Chisinau City MO, 45

Tags: locus , circles , equal segments , tangent

Determine the locus of points, from which the tangent segments to two given circles are equal.

1984 Bulgaria National Olympiad, Problem 6

Tags: geometry , 3d geometry , pyramid , parallelogram , locus

Let there be given a pyramid $SABCD$ whose base $ABCD$ is a parallelogram. Let $N$ be the midpoint of $BC$. A plane $\lambda$ intersects the lines $SC,SA,AB$ at points $P,Q,R$ respectively such that $\overline{CP}/\overline{CS}=\overline{SQ}/\overline{SA}=\overline{AR}/\overline{AB}$. A point $M$ on the line $SD$ is such that the line $MN$ is parallel to $\lambda$. Show that the locus of points $M$, when $\lambda$ takes all possible positions, is a segment of the length $\frac{\sqrt5}2SD$.

1983 Bundeswettbewerb Mathematik, 1

Tags: geometry , reflection , locus

The figure shows a triangular pool table with sides $a$, $b$ and $c$. Located at point $S$ on $c$ a sphere - which can be assumed as a point. After kick-off, as indicated in the figure, it runs through as a result of reflections to $a, b, a, b$ and $c$ (in $S$) always the same track. The reflection occurs according to law of reflection. Characterize entilrely all triangles $ABC$, which allow such an orbit, and determine the locus of $S$. [img]https://cdn.artofproblemsolving.com/attachments/5/b/7662943e5b9ad321226e0c5f5daa3c4ac9faaa.png[/img]

1978 Romania Team Selection Test, 5

Tags: geometry , pure geometry , locus

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

1956 Czech and Slovak Olympiad III A, 4

Tags: locus , locus of points , semicircle , geometry

Let a semicircle $AB$ be given and let $X$ be an inner point of the arc. Consider a point $Y$ on ray $XA$ such that $XY=XB$. Find the locus of all points $Y$ when $X$ moves on the arc $AB$ (excluding the endpoints).

Swiss NMO - geometry, 2008.5

Tags: geometry , locus , square

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

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.

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 $S.$

Ukrainian TYM Qualifying - geometry, XI.13

Tags: geometry , locus , 3d geometry , cylinder

On the plane there are two cylindrical towers with radii of bases $r$ and $R$. Find the set of all those points of the plane from which these towers are visible at the same angle. Consider the case of more towers.

1974 Czech and Slovak Olympiad III A, 6

Tags: geometry , geometric transformation , rotation , locus , square , area

Let a unit square $\mathcal D$ be given in the plane. For any point $X$ in the plane denote $\mathcal D_X$ the image of $\mathcal D$ in rotation with respect to origin $X$ by $+90^\circ.$ Find the locus of all $X$ such that the area of union $\mathcal D\cup\mathcal D_X$ is at most 1.5.

1969 IMO Longlists, 39

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

$(HUN 6)$ Find the positions of three points $A,B,C$ on the boundary of a unit cube such that $min\{AB,AC,BC\}$ is the greatest possible.

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 $AB$.

1981 Czech and Slovak Olympiad III A, 3

Tags: geometry , locus , square , equilateral , triangle

Let $ABCD$ be a unit square. Consider an equilateral triangle $XYZ$ with $X,Y$ as (inner or boundary) points of the square. Determine the locus $M$ of vertices $Z$ of all these triangles $XYZ$ and compute the area of $M.$

1989 Greece National Olympiad, 2

Tags: geometry , locus

Let $M$ be a point on side $BC$ of isosceles $ABC$ ($AB=AC$) and let $N$ be a points on the extension of $BC$ such that $(AM)^2+(AN)^2=2(AB)^2$. Find the locus of point $N$ when point $M$ moves on side $BC$.

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.

2000 Regional Competition For Advanced Students, 3

Tags: geometry , locus , midpoint

We consider two circles $k_1(M_1, r_1)$ and $k_2(M_2, r_2)$ with $z = M_1M_2 > r_1+r_2$ and a common outer tangent with the tangent points $P_1$ and $P2$ (that is, they lie on the same side of the connecting line $M_1M_2$). We now change the radii so that their sum is $r_1+r_2 = c$ remains constant. What set of points does the midpoint of the tangent segment $P_1P_2$ run through, when $r_1$ varies from $0$ to $c$?

2019 Argentina National Olympiad Level 2, 3

Tags: circumcenter , locus , circles , geometry

Let $\Gamma$ be a circle of center $S$ and radius $r$ and let be $A$ a point outside the circle. Let $BC$ be a diameter of $\Gamma$ such that $B$ does not belong to the line $AS$ and consider the point $O$ where the perpendicular bisectors of triangle $ABC$ intersect, that is, the circumcenter of $ABC$. Determine all possible locations of point $O$ when $B$ varies in circle $\Gamma$.