Found problems: 94
2024 Middle European Mathematical Olympiad, 3
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 $T$.
2004 Germany Team Selection Test, 2
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 $\Gamma$.
1973 IMO Shortlist, 17
$G$ 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.
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
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
1985 IMO Shortlist, 17
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 $n \geq 1.$
2006 Victor Vâlcovici, 1
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.
2006 Sharygin Geometry Olympiad, 2
Points $A, B$ move with equal speeds along two equal circles.
Prove that the perpendicular bisector of $AB$ passes through a fixed point.
2016 Vietnam Team Selection Test, 3
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.
Kyiv City MO Seniors 2003+ geometry, 2007.11.5
The points $A$ and $P$ are marked on the plane. Consider all such points $B, C $ of this plane that $\angle ABP = \angle MAB$ and $\angle ACP = \angle MAC $, where $M$ is the midpoint of the segment $BC$. Prove that all the circumscribed circles around the triangle $ABC$ for different points $B$ and $C$ pass through some fixed point other than the point $A$.
(Alexei Klurman)
2023 4th Memorial "Aleksandar Blazhevski-Cane", P2
Let $\mathbb{R}^{+}$ be the set of positive real numbers. Find all functions $f:\mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that for all $x,y>0$ we have
$$f(xy+f(x))=yf(x)+x.$$
[i]Proposed by Nikola Velov[/i]
2020 Dutch IMO TST, 2
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 $D$.
1979 IMO, 3
Two circles in a plane intersect. $A$ is one of the points of intersection. Starting simultaneously from $A$ two points move with constant speed, each travelling along its own circle in the same sense. The two points return to $A$ simultaneously after one revolution. Prove that there is a fixed point $P$ in the plane such that the two points are always equidistant from $P.$
2021-IMOC, G4
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 $AC$.
1992 Tournament Of Towns, (347) 5
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)
2024 Brazil Undergrad MO, 2
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 \( j, k \geq 2 \).
2019 Tournament Of Towns, 3
Two not necessarily equal non-intersecting wooden disks, one gray and one black, are glued to a plane. An in finite angle with one gray side and one black side can be moved along the plane so that the disks remain outside the angle, while the colored sides of the angle are tangent to the disks of the same color (the tangency points are not the vertices). Prove that it is possible to draw a ray in the angle, starting from the vertex of the angle and such that no matter how the angle is positioned, the ray passes through some fixed point of the plane.
(Egor Bakaev, Ilya Bogdanov, Pavel Kozhevnikov, Vladimir Rastorguev) (Junior version [url=https://artofproblemsolving.com/community/c6h2094701p15140671]here[/url])
[hide=note]There was a mistake in the text of the problem 3, we publish here the correct version. The solutions were estimated according to the text published originally.[/hide]
V Soros Olympiad 1998 - 99 (Russia), 10.10
A chord $AB$ is drawn in a circle. The line $\ell$ is parallel to $AB$ and does not intersect the circle. Let $C$ be a certain point on the circle (points $C$ located on one side of $AB$ are considered). Lines $CA$ and $CB$ intersect $\ell$ at points $D$ and $E$. Prove that there exists a fixed point $F$ of the plane, not lying on line $\ell$ , such that $\angle DFE$ is constant.
Swiss NMO - geometry, 2019.1
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 $k$.
2015 Oral Moscow Geometry Olympiad, 5
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.
1985 Traian Lălescu, 2.3
Let $ X $ be the power set of set of $ \{ 0\}\cup\mathbb{N} , $ and let be a function $ d:X^2\longrightarrow\mathbb{R} $ defined as
$$ d(U,V)=\sum_{n\in\mathbb{N}}\frac{\chi_U (n) +\chi_V (n) -2\chi_{U\cap V} (n)}{2} , $$
where $ \chi_W (n)=\left\{ \begin{matrix} 1,& n\in W\\ 0,& n\not\in W \end{matrix} \right. ,\quad\forall W\in X,\forall n\in\mathbb{N} . $
[b]a)[/b] Prove that there exists an unique $ V' $ such that $ \lim_{k\to\infty} d\left( \{ k+i|i\in\mathbb{N}\} , V'\right) =0. $
[b]b)[/b] Demonstrate that for all $ V\in X $ there exists a $ v\in\mathbb{N} $ with $ d\left( \left\{ \frac{3}{2} -\frac{1}{2}(-1)^{v} \right\} , V \right) >\frac{1}{k} . $
[b]c)[/b] Let $ f: X\longrightarrow X,\quad f(X)=\left\{ 1+x|x\in X\right\} . $ Calculate $ d\left( f(A),f(B) \right) $ in terms of $ d(A,B) $ and prove that $ f $ admits an unique fixed point.
2018 Polish MO Finals, 2
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 $f(X)=X$.
1985 IMO, 6
For every real number $x_1$, construct the sequence $x_1,x_2,\ldots$ by setting: \[ x_{n+1}=x_n(x_n+{1\over n}). \] Prove that there exists exactly one value of $x_1$ which gives $0<x_n<x_{n+1}<1$ for all $n$.
2025 ISI Entrance UGB, 1
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 $x_0 \in \mathbb{R}$, $f \left( x_0 \right) = x_0$.
2023 Stars of Mathematics, 3
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]