Given a circle with center $O$ and radius equal to $1$. $AB$ and $AC$ are the tangents to this circle from point $A$. Point $M$ on the circle is such that the areas of quadrilaterals $OBMC$ and $ABMC$ are equal. Find $MA$.

A convex, closed figure lies inside a given circle. The figure is seen from every point of the circumference at a right angle (that is, the two rays drawn from the point and supporting the convex figure are perpendicular). Prove that the center of the circle is a center of symmetry of the figure.

Let skew lines $PM, QN$ be given such that $PM\perp PQ\perp QN$. Let a plane $\sigma\perp PQ$ containing the midpoint $O$ of segment $PQ$ be given and in it a circle $k$ with center $O$ and given radius $r$. Consider all segments $XY$ with endpoint $X, Y$ on lines $PM, QN$, respectively, which contain a point of $k$. Show that segments $XY$ have the same length. Find the locus of all such points $X$.

Consider a triangle $ABC$, inscribed in $O$ and $\angle A < \angle B$. Some point $P$ outside the circle satisfies $$\angle A=\angle PBA =180^{\circ}- \angle PCB$$ Let $D$ be the intersection of line $PB$ and $O$(different from $B$), and $Q$ the intersection of the tangent line of $O$ passing through $A$ and line $CD$. Show that $CQ : AB=AQ^2:AD^2$.

* What is the radius of the largest possible circle inscribed into a cube with side $a$?

Several black and white checkers (tokens?) are standing along the circumference. Two men remove checkers in turn. The first removes all the black ones that had at least one white neighbour, and the second -- all the white ones that had at least one black neighbour. They stop when all the checkers are of the same colour. a) Let there be $40$ checkers initially. Is it possible that after two moves of each man there will remain only one (checker)? b) Let there be $1000$ checkers initially. What is the minimal possible number of moves to reach the position when there will remain only one (checker)?

Let $\omega$ be the semicircle with diameter $PQ$. A circle $k$ is tangent internally to $\omega$ and to segment $PQ$ at $C$. Let $AB$ be the tangent to $K$ perpendicular to $PQ$, with $A$ on $\omega$ and $B$ on the segment $CQ$. Show that $AC$ bisects angle $\angle PAB$

Consider a circle with center $O$ and radius $R,$ and let $A$ and $B$ be two points in the plane of this circle. [b]a.)[/b] Draw a chord $CD$ of the circle such that $CD$ is parallel to $AB,$ and the point of the intersection $P$ of the lines $AC$ and $BD$ lies on the circle. [b]b.)[/b] Show that generally, one gets two possible points $P$ ($P_{1}$ and $P_{2}$) satisfying the condition of the above problem, and compute the distance between these two points, if the lengths $OA=a,$ $OB=b$ and $AB=d$ are given.

Let $100$ discs lie on the plane in such a way that each two of them have a common point. Prove that there exists a point lying inside at least $15$ of these discs. (M. Kharitonov, A. Polyansky)

a) We are given $n$ circles $O_1, O_2, . . . , O_n$, passing through one point $O$. Let $A_1, . . . , A_n$ denote the second intersection points of $O_1$ with $O_2, O_2$ with $O_3$, etc., $O_n$ with $O_1$, respectively. We choose an arbitrary point $B_1$ on $O_1$ and draw a line segment through $A_1$ and $B_1$ to the second intersection with $O_2$ at $B_2$, then draw a line segment through $A_2$ and $B_2$ to the second intersection with $O_3$ at $B_3$, etc., until we get a point $B_n$ on $O_n$. We draw the line segment through $B_n$ and $A_n$ to the second intersection with $O_1$ at $B_{n+1}$. If $B_k$ and $A_k$ coincide for some $k$, we draw the tangent to $O_k$ through $A_k$ until this tangent intersects $O_{k+1}$ at $B_{k+1}$. Prove that $B_{n+1}$ coincides with $B_1$. b) for $n=3$ the same problem.

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.

$100$ red points divide a blue circle into $100$ arcs such that their lengths are all positive integers from $1$ to $100$ in an arbitrary order. Prove that there exist two perpendicular chords with red endpoints.

Circles with centers $P, Q$ and $R$, having radii $1, 2$ and $3$, respectively, lie on the same side of line $l$ and are tangent to $l$ at $P', Q'$ and $R'$, respectively, with $Q'$ between $P'$ and $R'$. The circle with center $Q$ is externally tangent to each of the other two circles. What is the area of triangle $PQR$? $\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}$

$L(a)$ is the line connecting the points of the unit circle corresponding to the angles $a$ and $\pi - 2a$. Prove that if $a + b + c = 2\pi$, then the lines $L (a), L (b)$ and $L (c)$ intersect at one point.

We are given three points $A,B,C$ on a semicircle. The tangent lines at $A$ and $B$ to the semicircle meet the extension of the diameter at points $M,N$ respectively. The line passing through $A$ that is perpendicular to the diameter meets $NC$ at $R$, and the line passing through $B$ that is perpendicular to the diameter meets $MC$ at $S$. If the line $RS$ meets the extension of the diameter at $Z$, prove that $ZC$ is tangent to the semicircle.

Denote by $\Omega$ the circumcircle of the acute triangle $ABC$. Point $D$ is the midpoint of the arc $BC$ of $\Omega$ not containing $A$. Circle $\omega$ centered at $D$ is tangent to the segment $BC$ at point $E$. Tangents to the circle $\omega$ passing through point $A$ intersect line $BC$ at points $K$ and $L$ such that points $B, K, L, C$ lie on the line $BC$ in that order. Circle $\gamma_1$ is tangent to the segments $AL$ and $BL$ and to the circle $\Omega$ at point $M$. Circle $\gamma_2$ is tangent to the segments $AK$ and $CK$ and to the circle $\Omega$ at point $N$. Lines $KN$ and $LM$ intersect at point $P$. Prove that $\sphericalangle KAP = \sphericalangle EAL$.

A square with sides $1$ and four circles of radius $1$ considered each having a vertex of have the square as the center. Find area of the shaded part (see figure). [img]https://cdn.artofproblemsolving.com/attachments/b/6/6e28d94094d69bac13c2702853ac2c906a80a1.png[/img]

In the plane, consider a circle with center $S$ and radius $1.$ Let $ABC$ be an arbitrary triangle having this circle as its incircle, and assume that $SA\leq SB\leq SC.$ Find the locus of [b]a.)[/b] all vertices $A$ of such triangles; [b]b.)[/b] all vertices $B$ of such triangles; [b]c.)[/b] all vertices $C$ of such triangles.

Let $c(0, R)$ be a circle with diameter $AB$ and $C$ a point, on it different than $A$ and $B$ such that $\angle AOC > 90^o$. On the radius $OC$ we consider the point $K$ and the circle $(c_1)$ with center $K$ and radius $KC = R_1$. We draw the tangents $AD$ and $AE$ from $A$ to the circle $(c_1)$. Prove that the straight lines $AC, BK$ and $DE$ are concurrent

On straight line $\ell$ there are points $A$, $B$, $C$ and $D$, following in the indicated order: $AB = a$, $BC = b$, $CD = c$. Segments $AD$ and $BC$ serve as chords of two circles, and the sum of the angular values of the arcs of these circles located on one side of $\ell$ is equal to $360^o$. A third circle passes through $A$ and $B$, intersecting the first two at points $K$ and $M$. The straight line $KM$ intersects $\ell$ at point $E$. Find $AE$.

Let $I$ be the incenter of the $ABC$ triangle. The circumference that passes through $I$ and has center in $A$ intersects the circumscribed circumference of the $ABC$ triangle at points $M$ and $N$. Prove that the line $MN$ is tangent to the inscribed circle of the $ABC$ triangle.

In the rectangle $ABCD$ with $AB> BC$, the perpendicular bisecotr of $AC$ intersects the side $CD$ at point $E$. The circle with the center at point $E$ and the radius $AE$ intersects again the side $AB$ at point $F$. If point $O$ is the orthogonal projection of point $C$ on the line $EF$, prove that points $B, O$ and $D$ are collinear.

Circles $\omega_1$ and $\omega_2$ intersect at points $X$ and $Y$. Line $\ell$ is tangent to $\omega_1$ and $\omega_2$ at $A$ and $B$, respectively, with line $AB$ closer to point $X$ than to $Y$. Circle $\omega$ passes through $A$ and $B$ intersecting $\omega_1$ again at $D \neq A$ and intersecting $\omega_2$ again at $C \neq B$. The three points $C$, $Y$, $D$ are collinear, $XC = 67$, $XY = 47$, and $XD = 37$. Find $AB^2$.

In the plane, $\Gamma$ is a circle with centre $O$ and radius $r, P$ and $Q$ are distinct points on $\Gamma , A$ is a point outside $\Gamma , M$ and $N$ are the midpoints of $PQ$ and $AO$ respectively. Suppose$ OA = 2a$ and $\angle PAQ$ is a right angle. Find the length of $MN$ in terms of $r$ and $a$. Express your answer in its simplest form, and justify your answer.

Two circles $ w_1 $ and $ w_2 $ intersect at two points $ P $ and $ Q $. The common tangent to $ w_1 $ and $ w_2 $, which is closer to the point $ P $ than to $ Q $, touches these circles at $ A $ and $ B $, respectively. The tangent to $ w_1 $ at the point $ P $ intersects $ w_2 $ at the point $ E $ (different from $ P $), and the tangent to $ w_2 $ at the point $ P $ intersects $ w_1 $ at $ F $ (different from $ P $). Let $ H $ and $ K $ be points on the rays $ AF $ and $ BE $, respectively, such that $ AH = AP $ and $ BK = BP $. Prove that the points $ A $, $ H $, $ Q $, $ K $ and $ B $ lie on the same circle.