Found problems: 95
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]
Let $ABC$ be an equiangular triangle with circumcircle $\omega$. Let point $F\in AB$ and point $E\in AC$ so that $\angle ABE+\angle ACF=60^{\circ}$. The circumcircle of triangle $AFE$ intersects the circle $\omega$ in the point $D$. The halflines $DE$ and $DF$ intersect the line through $B$ and $C$ in the points $X$ and $Y$. Prove that the incenter of the triangle $DXY$ is independent of the choice of $E$ and $F$.
(The angles in the problem statement are not directed. It is assumed that $E$ and $F$ are chosen in such a way that the halflines $DE$ and $DF$ indeed intersect the line through $B$ and $C$.)
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]
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]
In the non-isosceles triangle $ABC$ consider the points $X$ on $[AB]$ and $Y$ on $[AC]$ such that $[BX]=[CY]$, $M$ and $N$ are the midpoints of the segments $[BC]$, respectively $[XY]$, and the straight lines $XY$ and $BC$ meet in $K$. Prove that the circumcircle of triangle $KMN$ contains a point, different from $M$ , which is independent of the position of the points $X$ and $Y$.
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.
Given a triangle $ABC$, a line $\delta$ and a constant $k$, distinct from $0$ and $1,M$ a variable point on the line $\delta$. Points $E, F$ are on $MB,MC$ respectively such that $\frac{\overline{ME}}{\overline{MB}} = \frac{\overline{MF}}{\overline{MC}} = k$. Points $P,Q$ are on $AB,AC$ such that $PE, QF$ are perpendicular to $\delta$. Prove that the line through $M$ perpendicular to $PQ$ has a fixed point.
Nguyễn Minh Hà
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 $P.$
Pick a point $C$ on a semicircle with diameter $AB$. Let $P$ and $Q$ be two points on segment $AB$ such that $AP= AC$ and $BQ= BC$. The point $O$ is the center of the circumscribed circle of triangle $CPQ$ and point $H$ is the orthocenter of triangle $CPQ$ . Prove that for all posible locations of point $C$, the line $OH$ is passing through a fixed point.
(Mykhailo Sydorenko)
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 $f(x)=x$.
$G$ 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.
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)
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).
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 \( j, k \geq 2 \).
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)
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.
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.
The vertices of $\Delta ABC$ lie on the graphics of the function $f(x)=x^2$ and its centroid is $M(1,7)$. Determine the greatest possible value of the area of $\Delta ABC$.
From point $A$ to circle $\omega$ tangent $AD$ and arbitrary a secant intersecting a circle at points $B$ and $C$ (B lies between points $A$ and $C$). Prove that the circle passing through points $C$ and $D$ and touching the straight line $BD$, passes through a fixed point (other than $D$).
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 $P.$
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 $AMB$.
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)
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.
Two equal non-intersecting wooden disks, one gray and one black, are glued to a plane. A triangle with one gray side and one black side can be moved along the plane so that the disks remain outside the triangle, while the colored sides of the triangle are tangent to the disks of the same color (the tangency points are not the vertices). Prove that the line that contains the bisector of the angle between the gray and black sides always passes through some fixed point of the plane.
(Egor Bakaev, Pavel Kozhevnikov, Vladimir Rastorguev) (Senior version[url=https://artofproblemsolving.com/community/c6h2102856p15209040] here[/url])
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 $\Gamma$.