Olympiad problems

2023 Iran Team Selection Test, 4

line $l$ through the point $A$ from triangle $ABC$ . Point $X$ is on line $l$.${\omega}_b$ and ${\omega}_c$ are circles that through points $X,A$ and respectively tanget to $AB$ adn $AC$. tangets from $B,C$ respectively to ${\omega}_b$ and ${\omega}_c$ meet them in $Y,Z$. Prove that by changing $X$, the circumcircle of the circle $XYZ$ passes through two fixed points. [i]Proposed by Ali Zamani [/i]

Kyiv City MO Seniors Round2 2010+ geometry, 2010.10.4

Tags: fixed , geometry , fixed point , length , circumcircle , angle

The points $A \ne B$ are given on the plane. The point $C$ moves along the plane in such a way that $\angle ACB = \alpha$ , where $\alpha$ is the fixed angle from the interval ($0^o, 180^o$). The circle inscribed in triangle $ABC$ has center the point $I$ and touches the sides $AB, BC, CA$ at points $D, E, F$ accordingly. Rays $AI$ and $BI$ intersect the line $EF$ at points $M$ and $N$, respectively. Show that: a) the segment $MN$ has a constant length, b) all circles circumscribed around triangle $DMN$ have a common point

2016 Vietnam Team Selection Test, 3

Tags: geometry , fixed point , fixed , circles , circumcircle

Let $ABC$ be triangle with circumcircle $(O)$ of fixed $BC$, $AB \ne AC$ and $BC$ not a diameter. Let $I$ be the incenter of the triangle $ABC$ and $D = AI \cap BC, E = BI \cap CA, F = CI \cap AB$. The circle passing through $D$ and tangent to $OA$ cuts for second time $(O)$ at $G$ ($G \ne A$). $GE, GF$ cut $(O)$ also at $M, N$ respectively. a) Let $H = BM \cap CN$. Prove that $AH$ goes through a fixed point. b) Suppose $BE, CF$ cut $(O)$ also at $L, K$ respectively and $AH \cap KL = P$. On $EF$ take $Q$ for $QP = QI$. Let $J$ be a point of the circimcircle of triangle $IBC$ so that $IJ \perp IQ$. Prove that the midpoint of $IJ$ belongs to a fixed circle.

2019 Tournament Of Towns, 4

Tags: fixed , fixed point , isosceles , geometry

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)

Swiss NMO - geometry, 2019.1

Tags: geometry , fixed point , fixed , circles

Let $A$ be a point and let k be a circle through $A$. Let $B$ and $C$ be two more points on $k$. Let $X$ be the intersection of the bisector of $\angle ABC$ with $k$. Let $Y$ be the reflection of $A$ wrt point $X$, and $D$ the intersection of the straight line $YC$ with $k$. Prove that point $D$ is independent of the choice of $B$ and $C$ on the circle $k$.

2018 Polish MO Finals, 2

Tags: combinatorics , number theory , fixed point

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 $f(X)=X$.

2014 Hanoi Open Mathematics Competitions, 14

Tags: geometry , fixed point , circles

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 $P$.

2007 Sharygin Geometry Olympiad, 13

Tags: geometry , fixed point , circumcircle , equal segments

On the side $AB$ of a triangle $ABC$, two points $X, Y$ are chosen so that $AX = BY$. Lines $CX$ and $CY$ meet the circumcircle of the triangle, for the second time, at points $U$ and $V$. Prove that all lines $UV$ (for all $X, Y$, given $A, B, C$) have a common point.

2020 Dutch IMO TST, 2

Tags: geometry , reflection , fixed point , fixed , circumcircle , angle bisector

Given is a triangle $ABC$ with its circumscribed circle and $| AC | <| AB |$. On the short arc $AC$, there is a variable point $D\ne A$. Let $E$ be the reflection of $A$ wrt the inner bisector of $\angle BDC$. Prove that the line $DE$ passes through a fixed point, regardless of point $D$.

2005 Iran MO (3rd Round), 6

Tags: function , topology , algebra , fixed point

Suppose $A\subseteq \mathbb R^m$ is closed and non-empty. Let $f:A\to A$ is a lipchitz function with constant less than 1. (ie there exist $c<1$ that $|f(x)-f(y)|<c|x-y|,\ \forall x,y \in A)$. Prove that there exists a unique point $x\in A$ such that $f(x)=x$.

1998 IMC, 3

Tags: fixed point , real analysis

Given $ 0< c< 1$, we define $f(x) = \begin{cases} \frac{x}{c} & x \in [0,c] \\ \frac{1-x}{1-c} & x \in [c, 1] \end{cases} $ Let $f^{n}(x)=f(f(...f(x)))$ . Show that for each positive integer $n$, $f^{n}$ has a non-zero finite nunber of fixed points which aren't fixed points of $f^k$ for $k< n$.

2021-IMOC, G4

Tags: geometry , incenter , fixed point

Let $D$ be a point on the side $AC$ of a triangle $ABC$. Suppose that the incircle of triangle $BCD$ intersects $BD$ and $CD$ at $X$, $Y$, respectively. Show that $XY$ passes through a fixed point when $D$ is moving on the side $AC$.

2006 Victor Vâlcovici, 1

Tags: function , lipschitz , fixed point , short map , nonexpansive map , composition , real analysis

Let be a nondegenerate and closed interval $ I $ of real numbers, a short map $ m:I\longrightarrow I, $ and a sequence of functions $ \left( x_n \right)_{n\ge 1} :I\longrightarrow\mathbb{R} $ such that $ x_1 $ is the identity map and $$ 2x_{n+1}=x_n+m\circ x_n , $$ for any natural numbers $ n. $ Prove that: [b]a)[/b] there exists a nondegenerate interval having the property that any point of it is a fixed point for $ m. $ [b]b)[/b] $ \left( x_n \right)_{n\ge 1} $ is pointwise convergent and that its limit function is a short map.

2025 ISI Entrance UGB, 1

Tags: calculus , fixed point

Suppose $f \colon \mathbb{R} \longrightarrow \mathbb{R}$ is differentiable and $| f'(x)| < \frac{1}{2}$ for all $x \in \mathbb{R}$. Show that for some $x_0 \in \mathbb{R}$, $f \left( x_0 \right) = x_0$.

2015 Oral Moscow Geometry Olympiad, 5

Tags: geometry , 3d geometry , sphere , fixed point , moscow , plane

A triangle $ABC$ and spheres are given in space $S_1$ and $S_2$, each of which passes through points $A, B$ and $C$. For points $M$ spheres $S_1$ not lying in the plane of triangle $ABC$ are drawn lines $MA, MB$ and $MC$, intersecting the sphere $S_2$ for the second time at points $A_1,B_1$ and $C_1$, respectively. Prove that the planes passing through points $A_1, B_1$ and $C_1$, touch a fixed sphere or pass through a fixed point.

1992 Tournament Of Towns, (347) 5

Tags: geometry , fixed , fixed point

An angle with vertex $O$ and a point $A$ inside it are placed on a plane. Points $M$ and $N$ are chosen on different sides of the angle so that the angles $CAM$ and $CAN$ are equal. Prove that the straight line $MN$ always passes through a fixed point (or is always parallel to a fixed line). (S Tokarev)

2017 Balkan MO Shortlist, G1

Tags: geometry , circumcircle , fixed point

Let $ABC$ be an acute triangle. Variable points $E$ and $F$ are on sides $AC$ and $AB$ respectively such that $BC^2 = BA\cdot BF + CE \cdot CA$ . As $E$ and $F$ vary prove that the circumcircle of $AEF$ passes through a fixed point other than $A$ .

1986 IMO Shortlist, 17

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.

2022 Yasinsky Geometry Olympiad, 6

Tags: geometry , incenter , fixed point , fixed , circumcircle

Let $s$ be an arbitrary straight line passing through the incenter $I$ of the triangle $ABC$ . Line $s$ intersects lines $AB$ and $BC$ at points $D$ and $E$, respectively. Points $P$ and $Q$ are the centers of the circumscribed circles of triangles $DAI$ and $CEI$, respectively, and point $F$ is the second intersection point of these circles. Prove that the circumcircle of the triangle $PQF$ is always passes through a fixed point on the plane regardless of the position of the straight line $s$. (Matvii Kurskyi)

1991 Tournament Of Towns, (295) 2

Tags: fixed , fixed point , geometry

The chord $MN$ on the circle is fixed. For every diameter $AB$ of the circle consider the intersection point $C$ of the lines $AM$ and $BN$ and construct the line $\ell$ passing through $C$ perpendicularly to $AB$. Prove that all the lines $\ell$ pass through a fixed point. (E. Kulanin, Moscow)