Olympiad problems

2023 Myanmar IMO Training, 3

Let $\triangle ABC$ be a triangle such that $AB = AC$, and let its circumcircle be $\Gamma$. Let $\omega$ be a circle which is tangent to $AB$ and $AC$ at $B$ and $C$. Point $P$ belongs to $\omega$, and lines $PB$ and $PC$ intersect $\Gamma$ again at $Q$ and $R$. $X$ and $Y$ are points on lines $BR$ and $CQ$ such that $AX = XB$ and $AY = YC$. Show that as $P$ varies on $\omega$, the circumcircle of $\triangle AXY$ passes through a fixed point other than $A$.

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 $k$.

2025 Bulgarian Winter Tournament, 12.2

Tags: geometry , tangency , fixed point

In the plane are fixed two internally tangent circles $\omega$ and $\Omega$, so that $\omega$ is inside $\Omega$. Denote their common point by $T$. The point $A \neq T$ moves on $\Omega$ and point $B$ on $\Omega$ is such that $AB$ is tangent to $\omega$. The line through $B$, perpendicular to $AB$, meets the external angle bisector of $\angle ATB$ at $P$. Prove that, as $A$ varies on $\Omega$, the line $AP$ passes through a fixed point.

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 $P$.

2024 Brazil Undergrad MO, 2

Tags: real analysis , derivative , fixed point , algebra , polynomial

For each pair of integers $ j, k \geq 2 $, define the function $ f_{jk} : \mathbb{R} \to \mathbb{R} $ given by \[ f_{jk}(x) = 1 - (1 - x^j)^k. \] (a) Prove that for any integers $ j, k \geq 2 $, there exists a unique real number $ p_{jk} \in (0, 1) $ such that $ f_{jk}(p_{jk}) = p_{jk} $. Furthermore, defining $ \lambda_{jk} := f'_{jk}(p_{jk}) $, prove that $ \lambda_{jk} > 1 $. (b) Prove that $ p^j_{jk} = 1 - p_{kj} $ for any integers $ j, k \geq 2 $. (c) Prove that $ \lambda_{jk} = \lambda_{kj} $ for any integers $ j, k \geq 2 $.

2010 Oral Moscow Geometry Olympiad, 3

Tags: geometry , fixed , fixed point , area of a triangle , area

On the sides $AB$ and $BC$ of triangle $ABC$, points $M$ and $K$ are taken, respectively, so that $S_{KMC} + S_{KAC}=S_{ABC}$. Prove that all such lines $MK$ pass through one point.

2013 Bogdan Stan, 1

Tags: group theory , abstract algebra , algebra , polynomial , fixed point

Under composition, let be a group of linear polynomials that admit a fixed point . Show that all polynomials of this group have the same fixed point. [i]Vasile Pop[/i]

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)

1985 IMO Shortlist, 17

Tags: function , recurrence relation , fixed point , calculus , limit

The sequence $f_1, f_2, \cdots, f_n, \cdots $ of functions is defined for $x > 0$ recursively by \[f_1(x)=x , \quad f_{n+1}(x) = f_n(x) \left(f_n(x) + \frac 1n \right)\] Prove that there exists one and only one positive number $a$ such that $0 < f_n(a) < f_{n+1}(a) < 1$ for all integers $n \geq 1.$

1973 IMO, 2

Tags: function , algebra , fixed point , group theory

$G$ is a set of non-constant functions $f$. Each $f$ is defined on the real line and has the form $f(x)=ax+b$ for some real $a,b$. If $f$ and $g$ are in $G$, then so is $fg$, where $fg$ is defined by $fg(x)=f(g(x))$. If $f$ is in $G$, then so is the inverse $f^{-1}$. If $f(x)=ax+b$, then $f^{-1}(x)= \frac{x-b}{a}$. Every $f$ in $G$ has a fixed point (in other words we can find $x_f$ such that $f(x_f)=x_f$. Prove that all the functions in $G$ have a common fixed point.

2024 Middle European Mathematical Olympiad, 3

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

Let $ABC$ be an acute scalene triangle. Choose a circle $\omega$ passing through $B$ and $C$ which intersects the segments $AB$ and $AC$ at the interior points $D$ and $E$, respectively. The lines $BE$ and $CD$ intersects at $F$. Let $G$ be a point on the circumcircle of $ABF$ such that $GB$ is tangent to $\omega$ and let $H$ be a point on the circumcircle of $ACF$ such that $HC$ is tangent to $\omega$. Prove that there exists a point $T\neq A$, independent of the choice of $\omega$, such that the circumcircle of triangle $AGH$ passes through $T$.

2021 Czech and Slovak Olympiad III A, 6

Tags: geometry , fixed point , fixed

An acute triangle $ABC$ is given. Let us denote $X$ for each of its inner points $X_a, X_b, X_c$ its images in axial symmetries sequentially along the lines $BC, CA, AB$. Prove that all $X_aX_bX_c$ triangles have a common interior point. (Josef Tkadlec)

1985 IMO Longlists, 78

Tags: function , recurrence relation , fixed point , calculus , limit

The sequence $f_1, f_2, \cdots, f_n, \cdots $ of functions is defined for $x > 0$ recursively by \[f_1(x)=x , \quad f_{n+1}(x) = f_n(x) \left(f_n(x) + \frac 1n \right)\] Prove that there exists one and only one positive number $a$ such that $0 < f_n(a) < f_{n+1}(a) < 1$ for all integers $n \geq 1.$

2008 USA Team Selection Test, 7

Tags: circumcircle , parallelogram , spiral similarity , geometry , fixed point

Let $ ABC$ be a triangle with $ G$ as its centroid. Let $ P$ be a variable point on segment $ BC$. Points $ Q$ and $ R$ lie on sides $ AC$ and $ AB$ respectively, such that $ PQ \parallel AB$ and $ PR \parallel AC$. Prove that, as $ P$ varies along segment $ BC$, the circumcircle of triangle $ AQR$ passes through a fixed point $ X$ such that $ \angle BAG = \angle CAX$.

1986 IMO Shortlist, 17

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

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.

1999 Kazakhstan National Olympiad, 7

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

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 $

2003 IMO Shortlist, 2

Tags: geometry , fixed point

Three distinct points $A$, $B$, and $C$ are fixed on a line in this order. Let $\Gamma$ be a circle passing through $A$ and $C$ whose center does not lie on the line $AC$. Denote by $P$ the intersection of the tangents to $\Gamma$ at $A$ and $C$. Suppose $\Gamma$ meets the segment $PB$ at $Q$. Prove that the intersection of the bisector of $\angle AQC$ and the line $AC$ does not depend on the choice of $\Gamma$.

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 $D$.

2009 Postal Coaching, 5

Tags: fixed point , fixed , circumcircle , geometry

A point $D$ is chosen in the interior of the side $BC$ of an acute triangle $ABC$, and another point $P$ in the interior of the segment $AD$, but not lying on the median through $C$. This median (through $C$) intersects the circumcircle of a triangle $CPD$ at $K(\ne C)$. Prove that the circumcircle of triangle $AKP$ always passes through a fixed point $M(\ne A)$ independent of the choices of the points $D$ and $P.$

2012 USA Team Selection Test, 1

Tags: circumcircle , trigonometry , length , spiral similarity , fixed point , geometry

In acute triangle $ABC$, $\angle{A}<\angle{B}$ and $\angle{A}<\angle{C}$. Let $P$ be a variable point on side $BC$. Points $D$ and $E$ lie on sides $AB$ and $AC$, respectively, such that $BP=PD$ and $CP=PE$. Prove that as $P$ moves along side $BC$, the circumcircle of triangle $ADE$ passes through a fixed point other than $A$.

IV Soros Olympiad 1997 - 98 (Russia), 9.11

Tags: geometry , fixed , fixed point

Given two circles intersecting at points $A$ and $B$. A certain circle touches the first at point $A$, intersects the second at point $M$ and intersects the straight line $AB$ at point $P$ ($M$ and $P$ are different from $B$). Prove that the straight line $MP$ passes through a fixed point of the plane (for any change in the third circle).

2015 Sharygin Geometry Olympiad, 4

Tags: geometry , fixed point

A fixed triangle $ABC$ is given. Point $P$ moves on its circumcircle so that segments $BC$ and $AP$ intersect. Line $AP$ divides triangle $BPC$ into two triangles with incenters $I_1$ and $I_2$. Line $I_1I_2$ meets $BC$ at point $Z$. Prove that all lines $ZP$ pass through a fixed point. (R. Krutovsky, A. Yakubov)

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)

1986 Dutch Mathematical Olympiad, 4

Tags: geometry , fixed point , fixed , fixed circle

The lines $a$ and $b$ are parallel and the point $A$ lies on $a$. One chooses one circle $\gamma$ through A tangent to $b$ at $B$. $a$ intersects $\gamma$ for the second time at $T$. The tangent line at $T$ of $\gamma$ is called $t$. Prove that independently of the choice of $\gamma$, there is a fixed point $P$ such that $BT$ passes through $P$. Prove that independently of the choice of $\gamma$, there is a fixed circle $\delta$ such that $t$ is tangent to $\delta$.