Let $A_n$ be the area outside a regular $n$-gon of side length $1$ but inside its circumscribed circle, let $B_n$ be the area inside the $n$-gon but outside its inscribed circle. Find the limit as $n$ tends to infinity of $\dfrac{A_n}{B_n}$.

Let $ABC$ be an acute triangle, $k$ its circumcirle and $m$ a line such that $m\cap k=\emptyset, m\parallel BC.$ Denote $D$ the intersection of $m$ and ray $AB.$ a) Let $X$ be an inner point of the arc $BC$ not containing $A$ and denote $Y$ the intersection of lines $m,CX.$ Show that $A,D,X,Y$ are concyclic and name this circle $\kappa$. b) Determine relative position of $\kappa$ and $m$ in case when $C,D,X$ are collinear.

One of two circumferences of radius $R$ comes through $A$ and $B$ vertices of the $ABCD$ parallelogram. Another comes through $B$ and $D$. Let $M$ be another point of circumferences intersection. Prove that the circle circumscribed around $AMD$ triangle has radius $R$.

Let $ ABC$ be a triangle and $ O$ its circumcenter. Lines $ AB$ and $ AC$ meet the circumcircle of $ OBC$ again in $ B_1\neq B$ and $ C_1 \neq C$, respectively, lines $ BA$ and $ BC$ meet the circumcircle of $ OAC$ again in $ A_2\neq A$ and $ C_2\neq C$, respectively, and lines $ CA$ and $ CB$ meet the circumcircle of $ OAB$ in $ A_3\neq A$ and $ B_3\neq B$, respectively. Prove that lines $ A_2A_3$, $ B_1B_3$ and $ C_1C_2$ have a common point.

Given is the convex quadrilateral $ ABCD$. Assume that there exists a point $ P$ inside the quadrilateral for which the triangles $ ABP$ and $ CDP$ are both isosceles right triangles with the right angle at the common vertex $ P$. Prove that there exists a point $ Q$ for which the triangles $ BCQ$ and $ ADQ$ are also isosceles right triangles with the right angle at the common vertex $ Q$.

An acute-angled $ABC \ (AB<AC)$ is inscribed into a circle $\omega$. Let $M$ be the centroid of $ABC$, and let $AH$ be an altitude of this triangle. A ray $MH$ meets $\omega$ at $A'$. Prove that the circumcircle of the triangle $A'HB$ is tangent to $AB$. [i](A.I. Golovanov , A.Yakubov)[/i]

Let $ABC$ be a triangle such that $AB=6,BC=5,AC=7.$ Let the tangents to the circumcircle of $ABC$ at $B$ and $C$ meet at $X.$ Let $Z$ be a point on the circumcircle of $ABC.$ Let $Y$ be the foot of the perpendicular from $X$ to $CZ.$ Let $K$ be the intersection of the circumcircle of $BCY$ with line $AB.$ Given that $Y$ is on the interior of segment $CZ$ and $YZ=3CY,$ compute $AK.$

Let $\triangle{ABC}$ be a triangle, $\omega$ its circumcircle and $O$ the center of $\omega$. Let $P$ be a point on the segment $BC$. We denote by $Q$ the second intersection point of the circumcircles of triangles $\triangle{AOB}$ and $\triangle{APC}$. Prove that the line $PQ$ and the tangent to $\omega$ at point $A$ intersect on the circumcircle of triangle $\triangle AOB$.

Segment $AD$ is the diameter of the circumscribed circle of an acute-angled triangle $ABC$. Through the intersection of the altitudes of this triangle, a straight line was drawn parallel to the side $BC$, which intersects sides $AB$ and $AC$ at points $E$ and $F$, respectively. Prove that the perimeter of the triangle $DEF$ is two times larger than the side $BC$.

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.

Two circles $\Gamma_1$ and $\Gamma_2$ intersect at points $A$ and $Z$ (with $A\neq Z$). Let $B$ be the centre of $\Gamma_1$ and let $C$ be the centre of $\Gamma_2$. The exterior angle bisector of $\angle{BAC}$ intersects $\Gamma_1$ again at $X$ and $\Gamma_2$ again at $Y$. Prove that the interior angle bisector of $\angle{BZC}$ passes through the circumcenter of $\triangle{XYZ}$. [i]For points $P,Q,R$ that lie on a line $\ell$ in that order, and a point $S$ not on $\ell$, the interior angle bisector of $\angle{PQS}$ is the line that divides $\angle{PQS}$ into two equal angles, while the exterior angle bisector of $\angle{PQS}$ is the line that divides $\angle{RQS}$ into two equal angles.[/i]

In triangle $ABC$, the points $X$ and $Y$ are chosen on the arc $BC$ of the circumscribed circle of $ABC$ that doesn't contain $A$ so that $\measuredangle BAX = \measuredangle CAY$. Let $M$ be the midpoint of the segment $AX$. Show that $$BM + CM > AY.$$

Let $ABC$ be a triangle. A circle is tangent to sides $AB, AC$ and to the circumcircle of $ABC$ (internally) at points $P, Q, R$ respectively. Let $S$ be the point where $AR$ meets $PQ$. Show that \[\angle{SBA}\equiv \angle{SCA}\]

In acute triangle $ABC$, let $D,E,F$ denote the feet of the altitudes from $A,B,C$, respectively, and let $\omega$ be the circumcircle of $\triangle AEF$. Let $\omega_1$ and $\omega_2$ be the circles through $D$ tangent to $\omega$ at $E$ and $F$, respectively. Show that $\omega_1$ and $\omega_2$ meet at a point $P$ on $BC$ other than $D$. [i]Ray Li.[/i]

In an acute triangle $ABC$, altitudes at vertices $A$ and $B$ and bisector line at angle $C$ intersect the circumcircle again at points $A_1, B_1$ and $C_0$. Using the straightedge and compass, reconstruct the triangle by points $A_1, B_1$ and $C_0$.

Let $ K,L,M$ and $ N$ be the midpoints of sides $ AB,$ $ BC,$ $ CD$ and $ AD$ of the convex quadrangle $ ABCD.$ Is it possible that points $ A,B,L,M,D$ lie on the same circle and points $ K,B,C,D,N$ lie on the same circle?

In an isosceles triangle $ABC, I$ is the center of the inscribed circle, $M_1$ is the midpoint of the side $BC, K_2, K_3$ are the points of contact of the inscribed circle of the triangle with segments $AC$ and $AB$, respectively. The point $P$ lies on the circumcircle of the triangle $BCI$, and the angle $M_1PI$ is right. Prove that the lines $BC, PI, K_2K_3$ intersect at one point. (Mikhail Plotnikov)

Let $ABCD$ be a convex quadrilateral. Assume line $AB$ and $CD$ intersect at $E$, and $B$ lies between $A$ and $E$. Assume line $AD$ and $BC$ intersect at $F$, and $D$ lies between $A$ and $F$. Assume the circumcircles of $\triangle BEC$ and $\triangle CFD$ intersect at $C$ and $P$. Prove that $\angle BAP=\angle CAD$ if and only if $BD\parallel EF$.

An acute triangle $ABC$ ($AB > AC$) has circumcenter $O$, but $D$ is the midpoint of $BC$. Circle with diameter $AD$ intersects sides $AB$ and $AC$ in $E$ and $F$ respectively. On segment $EF$ pick a point $M$ so that $DM \parallel AO$. Prove that triangles $ABD$ and $FDM$ are similar.

(a) Triangles $A_1B_1C_1$ and $A_2B_2C_2$ are inscribed into triangle $ABC$ so that $C_1A_1 \perp BC$, $A_1B_1 \perp CA$, $B_1C_1 \perp AB$, $B_2A_2 \perp BC$, $C_2B_2 \perp CA$, $A_2C_2 \perp AB$. Prove that these triangles are equal. (b) Points $A_1$, $B_1$, $C_1$, $A_2$, $B_2$, $C_2$ lie inside a triangle $ABC$ so that $A_1$ is on segment $AB_1$, $B_1$ is on segment $BC_1$, $C_1$ is on segment $CA_1$, $A_2$ is on segment $AC_2$, $B_2$ is on segment $BA_2$, $C_2$ is on segment $CB_2$, and the angles $BAA_1$, $CBB_2$, $ACC_1$, $CAA_2$, $ABB_2$, $BCC_2$ are equal. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are equal.

Hexagon $ARTSCI$ has side lengths $AR=RT=TS=SC=4\sqrt2$ and $CI=IA=10\sqrt2$. Moreover, the vertices $A$, $R$, $T$, $S$, $C$, and $I$ lie on a circle $\mathcal{K}$. Find the area of $\mathcal{K}$.

Let $O$ be the circumcenter of $\triangle ABC$($AB<AC$), and $D$ be a point on the internal angle bisector of $\angle BAC$. Point $E$ lies on $BC$, satisfying $OE\parallel AD$, $DE\perp BC$. Point $K$ lies on $EB$ extended such that $EK=EA$. The circumcircle of $\triangle ADK$ meets $BC$ at $P\neq K$, and meets the circumcircle of $\triangle ABC$ at $Q\neq A$. Prove that $PQ$ is tangent to the circumcircle of $\triangle ABC$.

Let $O$ be the circumcenter,$R$ be the circumradius, and $k$ be the circumcircle of a triangle $ABC$ . Let $k_1$ be a circle tangent to the rays $AB$ and $AC$, and also internally tangent to $k$. Let $k_2$ be a circle tangent to the rays $AB$ and $AC$ , and also externally tangent to $k$. Let $A_1$ and $A_2$ denote the respective centers of $k_1$ and $k_2$. Prove that: $(OA_1+OA_2)^2-A_1A_2^2 = 4R^2.$

Let $ABCDE$ be a convex pentagon such that $BC \parallel AE,$ $AB = BC + AE,$ and $\angle ABC = \angle CDE.$ Let $M$ be the midpoint of $CE,$ and let $O$ be the circumcenter of triangle $BCD.$ Given that $\angle DMO = 90^{\circ},$ prove that $2 \angle BDA = \angle CDE.$ [i]Proposed by Nazar Serdyuk, Ukraine[/i]

Let $\ell$ be the perimeter of an acute-angled triangle $ABC$ which is not an equilateral triangle. Let $P$ be a variable points inside the triangle $ABC$, and let $D,E,F$ be the projections of $P$ on the sides $BC,CA,AB$ respectively. Prove that \[ 2(AF+BD+CE ) = \ell \] if and only if $P$ is collinear with the incenter and the circumcenter of the triangle $ABC$.