A triangle $ABC$ is given. Points $E,F,G$ are arbitrarily selected on the sides $AB,BC,CA$, respectively, such that $AF\perp EG$ and the quadrilateral $AEFG$ is cyclic. Find the locus of the intersection point of $AF$ and $EG$.

Which of the following four statements are true and which are false? a) If a polygon inscribed in a circle is equilateral, then it is also equiangular. b) If a polygon inscribed in a circle is equiangular, then it is also equilateral. c) If a polygon circumscribed to a circle is equilateral, then it is also equiangular. d) If a polygon circumscribed to a circle is equiangular, then it is also equilateral.

The trapezoid $ABCD$, of bases $AB$ and $CD$, is inscribed in a circumference $\Gamma$. Let $X$ a variable point of the arc $AB$ of $\Gamma$ that does not contain $C$ or $D$. We denote $Y$ to the point of intersection of $AB$ and $DX$, and let Z be the point of the segment $CX$ such that $\frac{XZ}{XC}=\frac{AY}{AB}$ . Prove that the measure of $\angle AZX$ does not depend on the choice of $X.$

Let $ABCD$ be an inscribed quadrilateral in a circle $c(O,R)$ (of circle $O$ and radius $R$). With centers the vertices $A,B,C,D$, we consider the circles $C_{A},C_{B},C_{C},C_{D}$ respectively, that do not intersect to each other . Circle $C_{A}$ intersects the sides of the quadrilateral at points $A_{1} , A_{2}$ , circle $C_{B}$ intersects the sides of the quadrilateral at points $B_{1} , B_{2}$ , circle $C_{C}$ at points $C_{1} , C_{2}$ and circle $C_{D}$ at points $C_{1} , C_{2}$ . Prove that the quadrilateral defined by lines $A_{1}A_{2} , B_{1}B_{2} , C_{1}C_{2} , D_{1}D_{2}$ is cyclic.

In the space $n \geq 3$ points are given. Every pair of points determines some distance. Suppose all distances are different. Connect every point with the nearest point. Prove that it is impossible to obtain (closed) polygonal line in such a way.

The side $AC$ of an acute triangle $ABC$ is the diameter of the circle $c_1$ and side $BC$ is the diameter of the circle $c_2$. Let $E$ be the foot of the altitude drawn from the vertex $B$ of the triangle and $F$ the foot of the altitude drawn from the vertex $A$. In addition, let $L$ and $N$ be the points of intersection of the line $BE$ with the circle $c_1$ (the point $L$ lies on the segment $BE$) and the points of intersection of $K$ and $M$ of line $AF$ with circle $c_2$ (point $K$ is in section $AF$). Prove that $K LM N$ is a cyclic quadrilateral.

Let $ABCDEF$ be a convex hexagon inscribed in a circle . Prove that the diagonals $AD, BE$ and $CF$ intersect at one point if and only if $$\frac{AB}{BC} \cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1$$

The quadrilateral $ABCD$ is known to be inscribed in a circle, and that there is a circle with center on side $AD$ tangent to the other three sides. Prove that $AD = AB + CD$.

The two circles $\Gamma_1$ and $\Gamma_2$ intersect at $P$ and $Q$. The common tangent that's on the same side as $P$, intersects the circles at $A$ and $B$,respectively. Let $C$ be the second intersection with $\Gamma_2$ of the tangent to $\Gamma_1$ at $P$, and let $D$ be the second intersection with $\Gamma_1$ of the tangent to $\Gamma_2$ at $Q$. Let $E$ be the intersection of $AP$ and $BC$, and let $F$ be the intersection of $BP$ and $AD$. Let $M$ be the image of $P$ under point reflection with respect to the midpoint of $AB$. Prove that $AMBEQF$ is a cyclic hexagon.

Different points $A_1,A_2,A_3,A_4,A_5$ lie on a circle so that $A_1A_2 = A_2A_3 = A_3A_4 =A_4A_5$. Let $A_6$ be the diametrically opposite point to $A_2$, and $A_7$ be the intersection of $A_1A_5$ and $A_3A_6$. Prove that the lines $A_1A_6$ and $A_4A_7$ are perpendicular

Let $AC>BC$ in $\triangle ABC$ and $M$ and $N$ be the midpoints of $AC$ and $BC$ respectively. The angle bisector of $\angle B$ intersects $\overline{MN}$ at $P$. The incircle of $\triangle ABC$ has center $I$ and touches $BC$ at $Q$. The perpendiculars from $P$ and $Q$ to $MN$ and $BC$ respectively intersect at $R$. Let $S=AB\cap RN$. a) Prove that $PCQI$ is cyclic b) Express the length of the segment $BS$ with $a$, $b$, $c$ - the side lengths of $\triangle ABC$ .

Is it possible to inscribe an $N$-gon in a circle so that all the lengths of its sides are different and all its angles (in degrees) are integer, where a) $N = 19$, b) $N = 20$ ? Mikhail Malkin

Let $ABCD$ be an inscribed quadrilateral. Denote the midpoints of the sides $AB, BC, CD$ and $DA$ through $M, L, N$ and $K$, respectively. It turned out that $\angle BM N = \angle MNC$. Prove that: i) $\angle DKL = \angle CLK$. ii) in the quadrilateral $ABCD$ there is a pair of parallel sides.

A convex hexagon $ABCDEF$ is given, with each two opposite sides of different lengths and parallel ($AB \parallel DE$, $BC \parallel EF$ and $CD \parallel FA$). If $|AE| = |BD|$ and $|BF| = |CE|$, prove that the hexagon $ABCDEF$ is cyclic.

A (convex) trapezoid $ABCD$ is good, if it is inscribed in a circle, sides $AB$ and $CD$ are the bases and $CD$ is shorter than $AB$. For a good trapezoid $ABCD$ the following terms are defined: $\bullet$ The parallel to $AD$ passing through $B$ intersects the extension of side $CD$ at point $S$. $\bullet$ The two tangents passing through $S$ on the circumircle of the trapezoid touch the circle at $E$ and $F$, where $E$ lies on the same side of the straight line $CD$ as $A$. Give the simplest possible equivalent condition (expressed in side lengths and / or angles of the trapezoid) so that with a good trapezoid $ABCD$ the two angles $\angle BSE$ and $\angle FSC$ have the same measure. (Walther Janous)

Pentagon $ABCDE$ is inscribed in a circle. Distances from point $E$ to lines $AB$ , $BC$ and $CD$ are equal to $a, b$ and $c$, respectively. Find the distance from point $E$ to line $AD$.

Let $A, B, C, D, E$ points in circle of radius r, in that order, such that $AC = BD = CE = r$. The points $H_1, H_2, H_3$ are the orthocenters of the triangles $ACD$, $BCD$ and $BCE$, respectively. Prove that $H_1H_2H_3$ is a right triangle .

Let $ABCD$ be a parallelogram and point $M$ be the midpoint of $[AB]$ so that the quadrilateral $MBCD$ is cyclic. If $N$ is the point of intersection of the lines $DM$ and $BC$, and $P \in BC$, then prove that the ray $(DP$ is the angle bisector of $\angle ADM$ if and only if $PC = 4BC$.

Let $A_1A_2... A_{2n + 1}$ be a convex polygon, $a_1 = A_1A_2$, $a_2 ​​= A_2A_3$, $...$, $a_{2n} = A_{2n}A_{2n + 1}$, $a_{2n + 1} = A_{2n + 1}A_1$. Denote by: $\alpha_i = \angle A_i$, $1 \le i \le 2n + 1$, $\alpha_{k + 2n + 1} = \alpha_k$, $k \ge 1$, $ \beta_i = \alpha_{i + 2} + \alpha_{i + 4} +... + \alpha_{i + 2n}$, $1 \le i \le 2n + 1$. Prove what if $$\frac{\alpha_1}{\sin \beta_1}=\frac{\alpha_2}{\sin \beta_2}=...=\frac{\alpha_{2n+1}}{\sin \beta_{2n+1}}$$ then a circle can be circumscribed around this polygon. Does the inverse statement hold a place?

Six different points $A, B, C, D, E, F$ are marked on the plane, no four of them lie on one circle and no two segments with ends at these points lie on parallel lines. Let $P, Q,R$ be the points of intersection of the perpendicular bisectors to pairs of segments $(AD, BE)$, $(BE, CF)$ ,$(CF, DA)$ respectively, and $P', Q' ,R'$ are points the intersection of the perpendicular bisectors to the pairs of segments $(AE, BD)$, $(BF, CE)$ , $(CA, DF)$ respectively. Show that $P \ne P', Q \ne Q', R \ne R'$, and prove that the lines $PP', QQ'$ and $RR'$ intersect at one point or are parallel.

The diagonals of some convex quadrilateral are mutually perpendicular and divide the quadrangle into $4$ triangles, the areas of which are expressed by prime numbers. Prove that a circle can be inscribed in this quadrilateral.

On the circle 6 distinct points $ A $, $ B $, $ C $, $ D $, $ E $, $ F $ are chosen in such a way that $ AB $ is parallel to $ DE $, and $ DC $ is parallel to $ AF $. Prove that $ BC $ is parallel to $ EF $

Let us call a convex hexagon $ABCDEF$ [i]boring [/i] if $\angle A+ \angle C + \angle E = \angle B + \angle D + \angle F$. a) Is every cyclic hexagon boring? b) Is every boring hexagon cyclic?

Let $ABCD$ be a quadrilateral whose diagonals are perpendicular, and let $S$ be the intersection of those diagonals. Let $K, L, M$ and $N$ be the reflections of $S$ on the sides $AB$, $BC$, $CD$ and $DA$ respectively. $BN$ cuts the circumcircle of $\vartriangle SKN$ at $E$ and $BM$ cuts the circumcircle of $\vartriangle SLM$ at $F$. Prove that the quadrilateral $EFLK$ is cyclic.

Let $ABCD$ be a convex quadrilateral with $\angle DAB=\angle BCD=90^{\circ}$ and $\angle ABC> \angle CDA$. Let $Q$ and $R$ be points on segments $BC$ and $CD$, respectively, such that line $QR$ intersects lines $AB$ and $AD$ at points $P$ and $S$, respectively. It is given that $PQ=RS$.Let the midpoint of $BD$ be $M$ and the midpoint of $QR$ be $N$.Prove that the points $M,N,A$ and $C$ lie on a circle.