In a triangle $ABC$ with $AB=AC$, $\omega$ is the circumcircle and $O$ its center. Let $D$ be a point on the extension of $BA$ beyond $A$. The circumcircle $\omega_{1}$ of triangle $OAD$ intersects the line $AC$ and the circle $\omega$ again at points $E$ and $G$, respectively. Point $H$ is such that $DAEH$ is a parallelogram. Line $EH$ meets circle $\omega_{1}$ again at point $J$. The line through $G$ perpendicular to $GB$ meets $\omega_{1}$ again at point $N$ and the line through $G$ perpendicular to $GJ$ meets $\omega$ again at point $L$. Prove that the points $L, N, H, G$ lie on a circle.

Let $ABC$ be an acute triangle and let $M$, $N$, and $ P$ be on $CB$, $AC$, and $AB$, respectively, such that $AB = AN + PB$, $BC = BP + MC$, $CA = CM + AN$. Let $\ell$ be a line in a different half plane than $C$ with respect to to the line $AB$ such that if $X, Y$ are the projections of $A, B$ on $\ell$ respectively, $AX = AP$ and $BY = BP$. Prove that $NXYM$ is a cyclic quadrilateral.

\item Two circles $ \Gamma_1$ and $ \Gamma_2$ intersect at points $ A$ and $ B$. Consider a circle $ \Gamma$ contained in $ \Gamma_1$ and $ \Gamma_2$, which is tangent to both of them at $ D$ and $ E$ respectively. Let $ C$ be one of the intersection points of line $ AB$ with $ \Gamma$, $ F$ be the intersection of line $ EC$ with $ \Gamma_2$ and $ G$ be the intersection of line $ DC$ with $ \Gamma_1$. Let $ H$ and $ I$ be the intersection points of line $ ED$ with $ \Gamma_1$ and $ \Gamma_2$ respectively. Prove that $ F$, $ G$, $ H$ and $ I$ are on the same circle.

Let $\Gamma_1$ and $\Gamma_2$ be two circles that tangents each other at point $N$, with $\Gamma_2$ located inside $\Gamma_1$. Let $A, B, C$ be distinct points on $\Gamma_1$ such that $AB$ and $AC$ tangents $\Gamma_2$ at $D$ and $E$, respectively. Line $ND$ cuts $\Gamma_1$ again at $K$, and line $CK$ intersects line $DE$ at $I$. (i) Prove that $CK$ is the angle bisector of $\angle ACB$. (ii) Prove that $IECN$ and $IBDN$ are cyclic quadrilaterals.

Circle $\omega$ is inscribed in quadrilateral $ABCD$ and is tangent to segments $BC, AD$ at $E,F$ , respectively.$DE$ intersects $\omega$ for the second time at $X$. if the circumcircle of triangle $DFX$ is tangent to lines $AB$ and $CD$ , prove that quadrilateral $AFXC$ is cyclic.

Points $P$ and $Q$ lie respectively on sides $AB$ and $AC$ of a triangle $ABC$ and $BP=CQ$. Segments $BQ$ and $CP$ cross at $R$. Circumscribed circles of triangles $BPR$ and $CQR$ cross again at point $S$ different from $R$. Prove that point $S$ lies on the bisector of angle $BAC$.

Let $ \text{ABCD} $ be a cyclic quadrilateral, with $ \text{AB} = 7 $, $ \text{BC} = 11 $, $ \text{CD} = 13 $, and $ \text{DA} = 17 $. Let the incircle of $ \text{ABD} $ hit $ \text{BD} $ at $ \text{R} $ and the incircle of $ \text{CBD} $ hit $ \text{BD} $ at $ \text{S} $. What is $ \text{RS} $?

On the side $ AB$ of a cyclic quadrilateral $ ABCD$ there is a point $ X$ such that diagonal $ BD$ bisects $ CX$ and diagonal $ AC$ bisects $ DX$. What is the minimum possible value of $ AB\over CD$? [i]Proposed by S. Berlov[/i]

Let $D$ be the point different from $B$ on the hypotenuse $AB$ of a right triangle $ABC$ such that $|CB| = |CD|$. Let $O$ be the circumcenter of triangle $ACD$. Rays $OD$ and $CB$ intersect at point $P$, and the line through point $O$ perpendicular to side AB and ray $CD$ intersect at point $Q$. Points $A, C, P, Q$ are concyclic. Does this imply that $ACPQ$ is a square?

Inside the inscribed quadrilateral $ ABCD $, a point $ P $ is marked such that $ \angle PBC = \angle PDA $, $ \angle PCB = \angle PAD $. Prove that there exists a circle that touches the straight lines $ AB $ and $ CD $, as well as the circles circumscribed by the triangles $ ABP $ and $ CDP $.

Given cyclic quadrilateral $ABCD$. Four circles each touching its diagonals and the circumcircle internally are equal. Is $ABCD$ a square? (C.Pohoata, A.Zaslavsky)

Let $W,X,Y$ and $Z$ be points on a circumference $\omega$ with center $O$, in that order, such that $WY$ is perpendicular to $XZ$; $T$ is their intersection. $ABCD$ is the convex quadrilateral such that $W,X,Y$ and $Z$ are the tangency points of $\omega$ with segments $AB,BC,CD$ and $DA$ respectively. The perpendicular lines to $OA$ and $OB$ through $A$ and $B$, respectively, intersect at $P$; the perpendicular lines to $OB$ and $OC$ through $B$ and $C$, respectively, intersect at $Q$, and the perpendicular lines to $OC$ and $OD$ through $C$ and $D$, respectively, intersect at $R$. $O_1$ is the circumcenter of triangle $PQR$. Prove that $T,O$ and $O_1$ are collinear. [i]Proposed by CDMX[/i]

Let $P$, $Q$, and $R$ be the points where the incircle of a triangle $ABC$ touches the sides $AB$, $BC$, and $CA$, respectively. Prove the inequality $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.

In scalene triangle $ABC$, let the feet of the perpendiculars from $A$ to $BC$, $B$ to $CA$, $C$ to $AB$ be $A_1, B_1, C_1$, respectively. Denote by $A_2$ the intersection of lines $BC$ and $B_1C_1$. Define $B_2$ and $C_2$ analogously. Let $D, E, F$ be the respective midpoints of sides $BC, CA, AB$. Show that the perpendiculars from $D$ to $AA_2$, $E$ to $BB_2$ and $F$ to $CC_2$ are concurrent.

Let $P$ be an arbitrary point inside triangle $ABC$, denote by $A_{1}$ (different from $P$) the second intersection of line $AP$ with the circumcircle of triangle $PBC$ and define $B_{1},C_{1}$ similarly. Prove that $\left(1 \plus{} 2\cdot\frac {PA}{PA_{1}}\right)\left(1 \plus{} 2\cdot\frac {PB}{PB_{1}}\right)\left(1 \plus{} 2\cdot\frac {PC}{PC_{1}}\right)\geq 8$.

Triangle $ABC$ has side lengths $AB=13$, $BC=14$, and $CA=15$. Points $D$ and $E$ are chosen on $AC$ and $AB$, respectively, such that quadrilateral $BCDE$ is cyclic and when the triangle is folded along segment $DE$, point $A$ lies on side $BC$. If the length of $DE$ can be expressed as $\tfrac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $100m+n$. [i]Proposd by Joseph Heerens[/i]

Consider a triangle $ABC$ with $AB < AC$ and let $H$ and $O$ be its orthocenter and circumcenter, respectively. A line starting from $B$ cuts the lines $AO$ and $AH$ at $M$ and $M'$ so that $M'$ is the midpoint of $BM$. Another line starting from $C$ cuts the lines $AH$ and $AO$ at $N$ and $N'$ so that $N'$ is the midpoint of $CN$. Prove that $M, M', N, N'$ are on the same circle.

Let $ABCD$ be a quadrilateral inscribed in the circle $(O)$. Let $(K)$ be an arbitrary circle passing through $B,C$. Circle $(O_1)$ tangent to $AB,AC$ and is internally tangent to $(K)$. Circle $(O_2)$ touches $DB,DC$ and is internally tangent to $(K)$. Prove that one of the two external common tangents of $(O_1)$ and $(O_2)$ is parallel to $AD$. Trần Quang Hùng

Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\omega$ whose diagonals meet at $F$. Lines $AB$ and $CD$ meet at $E$. Segment $EF$ intersects $\omega$ at $X$. Lines $BX$ and $CD$ meet at $M$, and lines $CX$ and $AB$ meet at $N$. Prove that $MN$ and $BC$ concur with the tangent to $\omega$ at $X$. [i]Proposed by Allen Liu[/i]

The quadrilateral $ABCD$ is inscribed in a circle centered at $O$, and $\{P\} = AC \cap BD, \{Q\} = AB \cap CD$. Let $R$ be the second intersection point of the circumcircles of the triangles $ABP$ and $CDP$. a) Prove that the points $P, Q$, and $R$ are collinear. b) If $U$ and $V$ are the circumcenters of the triangles $ABP$, and $CDP$, respectively, prove that the points $U, R, O, V$ are concyclic.