Olympiad problems

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 $X$.

2009 Mathcenter Contest, 2

Tags: geometry , locus , sq

Find the locus of points $P$ in the plane of a square $ABCD$ such that $$\max\{ PA,\ PC\}=\frac12(PB+PD).$$ [i](Anonymous314)[/i]

2012 Sharygin Geometry Olympiad, 4

Tags: locus , right triangle , square , geometry

Consider a square. Find the locus of midpoints of the hypothenuses of rightangled triangles with the vertices lying on three different sides of the square and not coinciding with its vertices. (B.Frenkin)

1969 IMO Longlists, 1

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

$(BEL 1)$ 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 $P_1.$

1965 IMO, 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 $\triangle OAB$.

2021/2022 Tournament of Towns, P4

Tags: geometry , 3d geometry , locus

Given is a segment $AB$. Three points $X, Y, Z$ are picked in the space so that $ABX$ is an equilateral triangle and $ABYZ$ is a square. Prove that the orthocenters of all triangles $XYZ$ obtained in this way belong to a fixed circle. [i]Alexandr Matveev[/i]

1963 Czech and Slovak Olympiad III A, 3

Tags: geometry , locus , common tangent , tangent circle

A line $MN$ is given in the plane. Consider circles $k_1$, $k_2$ tangent to the line at points $M$, $N$, respectively, while touching each other externally. Let $X$ be the midpoint of the segment $PQ$, where $P$, $Q$ are in this order tangent points of the second common external tangent of the circles $k_1$, $k_2$. Find the locus of the points $X$ for all pairs of circles of the specified properties.

1980 Poland - Second Round, 3

Tags: geometry , locus , 3d geometry , sphere

There is a sphere $ K $ in space and points $ A, B $ outside the sphere such that the segment $ AB $ intersects the interior of the sphere. Prove that the set of points $ P $ for which the segments $ AP $ and $ BP $ are tangent to the sphere $ K $ is contained in a certain plane.

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 $k$.

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) .

Kvant 2024, M2784

Tags: geometry , locus

The bisectors $AD{}$ and $BE{}$ were drawn in the triangle $ABC{}$ and they intersected at point $I{}.$ Then everything was erased, leaving only the points $D{}$ and $E{}.$ Find the set of possible positions of the point $I{}.$ [i]Proposed by M. Didin[/i]

1904 Eotvos Mathematical Competition, 3

Tags: geometry , geometric inequality , inequalities , rectangle , locus

Let $A_1A_2$ and $B_1B_2$ be the diagonals of a rectangle, and let $O$ be its center. Find and construct the set of all points $P$ that satisfy simultaneously the four inequaliies: $$A_1P > OP , \\A_2P > OP, \ \ B_1P > OP , \ \ B_2P > OP.$$

2005 Czech And Slovak Olympiad III A, 4

Tags: geometry , rectangle , locus , intersection diagonals

An acute-angled triangle $AKL$ is given on a plane. Consider all rectangles $ABCD$ circumscribed to triangle $AKL$ such that point $K$ lies on side $BC$ and point $L$ lieson side $CD$. Find the locus of the intersection $S$ of the diagonals $AC$ and $BD$.

1986 IMO Longlists, 47

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 $X$.

2012 Tournament of Towns, 4

Tags: geometry , cyclic , locus , circles , tangent circle

A quadrilateral $ABCD$ with no parallel sides is inscribed in a circle. Two circles, one passing through $A$ and $B$, and the other through $C$ and $D$, are tangent to each other at $X$. Prove that the locus of $X$ is a circle.

1957 Moscow Mathematical Olympiad, 351

Tags: geometry , locus , rectangle , concentric circles

Given two concentric circles and a pair of parallel lines. Find the locus of the fourth vertices of all rectangles with three vertices on the concentric circles, two vertices on one circle and the third on the other and with sides parallel to the given lines.

1969 IMO Shortlist, 53

Tags: trigonometry , geometry , locus , locus problems

$(POL 2)$ Given two segments $AB$ and $CD$ not in the same plane, find the locus of points $M$ such that $MA^2 +MB^2 = MC^2 +MD^2.$

II Soros Olympiad 1995 - 96 (Russia), 9.9

Tags: geometry , locus

Two points $A$ and $B$ are given on the plane. An arbitrary circle passes through $B$ and intersects the straight line $AB$ for second time at a point $K$, different from $A$. A circle passing through $A$, $K$ and the center of the first circle intersects the first one for second time at point $M$. Find the locus of points $M$.

2009 Romania National Olympiad, 1

Tags: geometry , locus problems , locus , vector geometry

On the sides $ AB,AC $ of a triangle $ ABC, $ consider the points $ M, $ respectively, $ N $ such that $ M\neq A\neq N $ and $ \frac{MB}{MA}\neq\frac{NC}{NA}. $ Show that the line $ MN $ passes through a point not dependent on $ M $ and $ N. $

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.

1963 IMO, 2

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

Point $A$ and segment $BC$ are given. Determine the locus of points in space which are vertices of right angles with one side passing through $A$, and the other side intersecting segment $BC$.

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 $L_1$.

2008 Cuba MO, 8

Tags: geometry , rectangle , locus

Let $ABC$ an acute-angle triangle. Let $R$ be a rectangle with vertices in the edges of $ABC$. Let $O$ be the center of $R$. a) Find the locus of all the points $O$. b) Decide if there is a point that is the center of three of these rectangles.

2021 Ecuador NMO (OMEC), 3

Tags: geometry , perimeter , locus

Let $T_1$ and $T_2$ internally tangent circumferences at $P$, with radius $R$ and $2R$, respectively. Find the locus traced by $P$ as $T_1$ rolls tangentially along the entire perimeter of $T_2$

2014 Hanoi Open Mathematics Competitions, 7

Tags: geometry , locus , tangent circle

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 $AB$.