Consider a cyclic quadrilateral such that the midpoints of its sides form another cyclic quadrilateral. Prove that the area of the smaller circle is less than or equal to half the area of the bigger circle

$100$ points at a circle with radius $1$ $cm$. Show that, we find an another point such that, this point's distance to other $100$ points is greater than $100$ $cm$.

In the trapezium $ABCD$, a point $M$ is chosen on the non-base segment $AB$. Through the points $M,A,D$ and $M,B,C$ are drawn circles $k_1$ and $k_2$ with centers $O_1$ and $O_2$. Prove that: (a) the second intersection point $N$ of $k_1$ and $k_2$ lies on the other non-base segment $CD$ or on its continuation; (b) the length of the line $O_1O_2$ doesn’t depend on the location of $M$ on $AB$; (c) the triangles $O_1MO_2$ and $DMC$ are similar. Find such a position of $M$ on $AB$ that makes $k_1$ and $k_2$ have the same radius.

Given a square whose side measures $a$, consider the set of all points of its plane through which passes a circumference of radius whose circle contains to the quoted square. You are asked to prove that the contour of the figure formed by the points with this property is formed by arcs of circumference, and determine the positions, their centers, their radii and their lengths.

The circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$, point $M$ is the midpoint of $AB$. On line $AB$ select points $S_1$ and $S_2$. Let $S_1X_1$ and $S_1Y_1$ be tangents drawn from $S_1$ to circle $\omega_1$, similarly $S_2X_2$ and $S_2Y_2$ are tangents drawn from $S_2$ to circles $\omega_2$. Prove that if the point $M$ lies on the line $X_1X_2$, then it also lies on the line $Y_1Y_2$.

Given a circle $K$, find the locus of vertices $A$ of parallelograms $ABCD$ with diagonals $AC \leq BD$, such that $BD$ is inside $K$.

The $ AB, BC $ and $ CD $ segments of the polygon $ ABCD $ have the same length and are tangent to a circle $ S $, centered on the point $ O $. Let $ P $ be the point of tangency of $ BC $ with $ S $, and let $ Q $ be the intersection point of lines $ AC $ and $ BD $. Show that the point $ Q $ is collinear with the points $ P $ and $ O $.

A polyline $AMB$ is inscribed in the arc $AB{}$, consisting of two segments, and $AM>MB$. Let $K$ be the midpoint of the arc $AB{}$. Prove that the foot $H{}$ of the perpendicular from $K$ onto $AM$ divides the polyline in two equal segments: \[AH=HM+MB.\][i]Discovered by Archimedes[/i]

Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$. Let the internal angle bisectors at $A$ and $B$ meet at $X$, the internal angle bisectors at $B$ and $C$ meet at $Y$, the internal angle bisectors at $C$ and $D$ meet at $Z$, and the internal angle bisectors at $D$ and $A$ meet at $W$. Further, let $AC$ and $BD$ meet at $P$. Suppose that the points $X$, $Y$, $Z$, $W$, $O$, and $P$ are distinct. Prove that $O$, $X$, $Y$, $Z$, $W$ lie on the same circle if and only if $P$, $X$, $Y$, $Z$, and $W$ lie on the same circle.

A circle $\omega$ touches the sides $A$B and $BC$ of a triangle $ABC$ and intersects its side $AC$ at $K$. It is known that the tangent to $\omega$ at $K$ is symmetrical to the line $AC$ with respect to the line $BK$. What can be the difference $AK -CK$ if $AB = 9$ and $BC = 11$?

For some values of c, the equation $x^c + y^c = z^c$ can be illustrated geometrically. For example, the case $c = 2$ can be illustrated by a right-angled triangle. By this we mean that, x, y, z is a solution of the equation $x^2 + y^2 = z^2$ if and only if there exists a right-angled triangle with catheters $x$ and $y$ and hypotenuse $z$. In this problem we will look at the cases $c = -\frac{1}{2}$ and $c = - 1$. a) Let $x, y$ and $z$ be the radii of three circles intersecting each other and a line, as shown, in the figure. Show that, $x^{-\frac{1}{2}}+ y^{-\frac{1}{2}} = z^{-\frac{1}{2}}$ [img]https://cdn.artofproblemsolving.com/attachments/5/7/5315e33e1750a3a49ae11e1b5527311117ce70.png[/img] b) Draw a geometric figure that illustrates the case in a similar way, $c = - 1$. The figure must be able to be constructed with a compass and a ruler. Describe such a construction and prove that, in the figure, lines $x, y$ and $z$ satisfy $x^{-1}+ y^{-1} = z^{-1}$. (All positive solutions of this equation should be possible values for $x, y$, and $z$ on such a figure, but you don't have to prove that.)

Two circles $\alpha,\beta$ touch externally at the point $X$. Let $A,P$ be two distinct points on $\alpha$ different from $X$, and let $AX$ and $PX$ meet $\beta$ again in the points $B$ and $Q$ respectively. Prove that $AP$ is parallel to $QB$.

A circle is divided by $2n$ points into $2n$ equal arcs. Let $P_1, P_2, \ldots, P_{2n}$ be an arbitrary permutation of the $2n$ division points. Prove that the polygonal line $P_1 P_2 \cdots P_{2n} P_1$ contains at least two parallel segments.

Let $\Gamma_1$ and $\Gamma_2$ be two intersecting circles with midpoints respectively $O_1$ and $O_2$, such that $\Gamma_2$ intersects the line segment $O_1O_2$ in a point $A$. The intersection points of $\Gamma_1$ and $\Gamma_2$ are $C$ and $D$. The line $AD$ intersects $\Gamma_1$ a second time in $S$. The line $CS$ intersects $O_1O_2$ in $F$. Let $\Gamma_3$ be the circumcircle of triangle $AD$. Let $E$ be the second intersection point of $\Gamma_1$ and $\Gamma_3$. Prove that $O_1E$ is tangent to $\Gamma_3$.

Let $ABCD$ be a parallelogram. An interior circle of the $ABCD$ is tangent to the lines $AB$ and $AD$ and intersects the diagonal $BD$ at $E$ and $F$. Prove that exists a circle that passes through $E$ and $F$ and is tangent to the lines $CB$ and $CD$.