Olympiad problems

Geometry Mathley 2011-12, 15.4

Let $ABC$ be a fixed triangle. Point $D$ is an arbitrary point on the side $BC$. Point $P$ is fixed on $AD$. The circumcircle of triangle $BPD$ meets $AB$ at $E$ distinct from $B$. Point $Q$ varies on $AP$. Let $BQ$ and $CQ$ meet the circumcircles of triangles $BPD, CPD$ respectively at $F,Z$ distinct from $B,C$. Prove that the circumcircle $EFZ$ is through a fixed point distinct from $E$ and this fixed point is on the circumcircle of triangle $CPD$. Kostas Vittas

2022 Adygea Teachers' Geometry Olympiad, 2

Tags: geometry , trapezoid , fixed , fixed point

An arbitrary point $P$ is chosen on the lateral side $AB$ of the trapezoid $ABCD$. Straight lines passing through it parallel to the diagonals of the trapezoid intersect the bases at points $Q$ and $R$. Prove that the sides $QR$ of all possible triangles $PQR$ pass through a fixed point.

2019 Tournament Of Towns, 2

Tags: circles , orthocenter , fixed point , fixed , geometry

Let $\omega$ be a circle with the center $O$ and $A$ and $C$ be two different points on $\omega$. For any third point $P$ of the circle let $X$ and $Y$ be the midpoints of the segments $AP$ and $CP$. Finally, let $H$ be the orthocenter (the point of intersection of the altitudes) of the triangle $OXY$ . Prove that the position of the point H does not depend on the choice of $P$. (Artemiy Sokolov)

2018 Polish MO Finals, 2

Tags: number theory , fixed point , combinatorics

A subset $S$ of size $n$ of a plane consisting of points with both coordinates integer is given, where $n$ is an odd number. The injective function $f\colon S\rightarrow S$ satisfies the following: for each pair of points $A, B\in S$, the distance between points $f(A)$ and $f(B)$ is not smaller than the distance between points $A$ and $B$. Prove there exists a point $X$ such that $f(X)=X$.

1999 Kazakhstan National Olympiad, 7

Tags: perpendicular , sphere , 3d geometry , area of triangle , fixed point , geometry

On a sphere with radius $1$, a point $ P $ is given. Three mutually perpendicular the rays emanating from the point $ P $ intersect the sphere at the points $ A $, $ B $ and $ C $. Prove that all such possible $ ABC $ planes pass through fixed point, and find the maximum possible area of the triangle $ ABC $

2019 Tournament Of Towns, 4

Tags: geometry , fixed , fixed point , isosceles

Isosceles triangles with a fixed angle $\alpha$ at the vertex opposite to the base are being inscribed into a rectangle $ABCD$ so that this vertex lies on the side $BC$ and the vertices of the base lie on the sides $AB$ and $CD$. Prove that the midpoints of the bases of all such triangles coincide. (Igor Zhizhilkin)

2005 Sharygin Geometry Olympiad, 11.5

Tags: geometry , fixed point , fixed , angle

The angle and the point $K$ inside it are given on the plane. Prove that there is a point $M$ with the following property: if an arbitrary line passing through intersects the sides of the angle at points $A$ and $B$, then $MK$ is the bisector of the angle $AMB$.

Kyiv City MO Juniors Round2 2010+ geometry, 2019.9.3

Tags: incenter , fixed , fixed point , equilateral , geometry

The equilateral triangle $ABC$ is inscribed in the circle $w$. Points $F$ and $E$ on the sides $AB$ and $AC$, respectively, are chosen such that $\angle ABE+ \angle ACF = 60^o$. The circumscribed circle of $\vartriangle AFE$ intersects the circle $w$ at the point $D$ for the second time. The rays $DE$ and $DF$ intersect the line $BC$ at the points $X$ and $Y$, respectively. Prove that the center of the inscribed circle of $\vartriangle DXY$ does not depend on the choice of points $F$ and $E$. (Hilko Danilo)

2017 German National Olympiad, 2

Tags: circumcircle , fixed point , geometry

Let $ABC$ be a triangle such that $\vert AB\vert \ne \vert AC\vert$. Prove that there exists a point $D \ne A$ on its circumcircle satisfying the following property: For any points $M, N$ outside the circumcircle on the rays $AB$ and $AC$, respectively, satisfying $\vert BM\vert=\vert CN\vert$, the circumcircle of $AMN$ passes through $D$.

1993 Poland - Second Round, 2

Tags: geometry , concurrency , fixed , fixed point

Let be given a circle with center $O$ and a point $P$ outside the circle. A line $l$ passes through $P$ and cuts the circle at $A$ and $B$. Let $C$ be the point symmetric to $A$ with respect to $OP$, and let $m$ be the line $BC$. Prove that all lines $m$ have a common point as $l$ varies.

2023 Stars of Mathematics, 3

Tags: geometry , fixed point

Let $ABC$ be an acute triangle, with $AB<AC{}$ and let $D$ be a variable point on the side $AB{}$. The parallel to $D{}$ through $BC{}$ crosses $AC{}$ at $E{}$. The perpendicular bisector of $DE{}$ crosses $BC{}$ at $F{}$. The circles $(BDF)$ and $(CEF)$ cross again at $K{}$. Prove that the line $FK{}$ passes through a fixed point. [i]Proposed by Ana Boiangiu[/i]

1998 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: geometry , fixed point , arc midpoint , arc , circles

Consider an arc $AB$ of a circle $C$ and a point $P$ variable in that arc $AB$. Let $D$ be the midpoint of the arc $AP$ that doeas not contain $B$ and let $E$ be the midpoint of the arc $BP$ that does not contain $A$. Let $C_1$ be the circle with center $D$ passing through $A$ and $C_2$ be the circle with center $E$ passing through $B.$ Prove that the line that contains the intersection points of $C_1$ and $C_2$ passes through a fixed point.

VI Soros Olympiad 1999 - 2000 (Russia), 10.10

Tags: geometry , fixed , fixed point

Take an arbitrary point $D$ on side $BC$ of triangle $ABC$ and draw a circle through point $D$ and the centers of the circles inscribed in triangles $ABD$ and $ACD$. Prove that all circles obtained for different points $D$ of side $BC$ have a common point.

2014 Hanoi Open Mathematics Competitions, 14

Tags: geometry , circles , fixed point

Let $\omega$ be a circle with centre $O$, and let $\ell$ be a line that does not intersect $\omega$. Let $P$ be an arbitrary point on $\ell$. Let $A,B$ denote the tangent points of the tangent lines from $P$. Prove that $AB$ passes through a point being independent of choosing $P$.

1992 Austrian-Polish Competition, 5

Tags: geometry , circles , fixed point , fixed , product

Given a circle $k$ with center $M$ and radius $r$, let $AB$ be a fixed diameter of $k$ and let $K$ be a fixed point on the segment $AM$. Denote by $t$ the tangent of $k$ at $A$. For any chord $CD$ through $K$ other than $AB$, denote by $P$ and Q the intersection points of $BC$ and $BD$ with $t$, respectively. Prove that $AP\cdot AQ$ does not depend on $CD$.

2012 Oral Moscow Geometry Olympiad, 5

Tags: geometry , circumcircle , orthocenter , fixed , fixed point

Given a circle and a chord $AB$, different from the diameter. Point $C$ moves along the large arc $AB$. The circle passing through passing through points $A, C$ and point $H$ of intersection of altitudes of of the triangle $ABC$, re-intersects the line $BC$ at point $P$. Prove that line $PH$ passes through a fixed point independent of the position of point $C$.

1986 IMO Longlists, 14

Tags: geometry , rotation , fixed point , equilateral triangle , triangle

Given a point $P_0$ in the plane of the triangle $A_1A_2A_3$. Define $A_s=A_{s-3}$ for all $s\ge4$. Construct a set of points $P_1,P_2,P_3,\ldots$ such that $P_{k+1}$ is the image of $P_k$ under a rotation center $A_{k+1}$ through an angle $120^o$ clockwise for $k=0,1,2,\ldots$. Prove that if $P_{1986}=P_0$, then the triangle $A_1A_2A_3$ is equilateral.

VMEO IV 2015, 11.2

Tags: geometry , fixed point , fixed , isogonal , projection

Let $ABC$ be a triangle with two isogonal points $ P$ and $Q$ . Let $D, E$ be the projection of $P$ on $AB$, $AC$. $G$ is the projection of $Q$ on $BC$. $U$ is the projection of $G$ on $DE$, $ L$ is the projection of $P$ on $AQ$, $K$ is the symmetric of $L$ wrt $UG$. Prove that $UK$ passes through a fixed point when $P$ and $Q$ vary.

1963 All Russian Mathematical Olympiad, 031

Tags: geometry , fixed point

Given two fixed points $A$ and $B$ .The point $M$ runs along the circumference containing $A$ and $B$. $K$ is the midpoint of the segment $[MB]$. $[KP]$ is a perpendicular to the line $(MA)$. a) Prove that all the possible lines $(KP)$ pass through one point. b) Find the set of all the possible points $P$.

2002 Chile National Olympiad, 3

Tags: geometry , fixed point

Given the line $AB$, let $M$ be a point on it. Towards the same side of the plane and with bases $AM$ and $MB$, squares $AMCD$ and $MBEF$ are constructed. Let $N$ be the point (different from $M$) where the circumcircles circumscribed to both squares intersect and let $N_1$ be the point where the lines $BC$ and $AF$ intersect. Prove that the points $N$ and $N_1$ coincide. Prove that as the point $M$ moves on the line $AB$, the line $MN$ moves always passing through a fixed point.