Given triangle $ABC$, the three altitudes intersect at point $H$. Determine all points $X$ on the side $BC$ so that the symmetric of $H$ wrt point $X$ lies on the circumcircle of triangle $ABC$.

Lines $a_1, b_1, c_1$ pass through the vertices $A, B, C$ respectively of a triange $ABC$; $a_2, b_2, c_2$ are the reflections of $a_1, b_1, c_1$ about the corresponding bisectors of $ABC$; $A_1 = b_1 \cap c_1, B_1 = a_1 \cap c_1, C_1 = a_1 \cap b_1$, and $A_2, B_2, C_2$ are defined similarly. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ have the same ratios of the area and circumradius (i.e. $\frac{S_1}{R_1} = \frac{S_2}{R_2}$, where $S_i = S(\triangle A_iB_iC_i)$, $R_i = R(\triangle A_iB_iC_i)$)

Let $\Gamma$ be a circle and let $d$ be a line such that $\Gamma$ and $d$ have no common points. Further, let $AB$ be a diameter of the circle $\Gamma$; assume that this diameter $AB$ is perpendicular to the line $d$, and the point $B$ is nearer to the line $d$ than the point $A$. Let $C$ be an arbitrary point on the circle $\Gamma$, different from the points $A$ and $B$. Let $D$ be the point of intersection of the lines $AC$ and $d$. One of the two tangents from the point $D$ to the circle $\Gamma$ touches this circle $\Gamma$ at a point $E$; hereby, we assume that the points $B$ and $E$ lie in the same halfplane with respect to the line $AC$. Denote by $F$ the point of intersection of the lines $BE$ and $d$. Let the line $AF$ intersect the circle $\Gamma$ at a point $G$, different from $A$. Prove that the reflection of the point $G$ in the line $AB$ lies on the line $CF$.

Let $ABC$ be a triangle inscribed in circle $\Gamma$ with center $O$. Let $H$ be the orthocenter of triangle $ABC$ and let $K$ be the midpoint of $OH$. Tangent of $\Gamma$ at $B$ intersects the perpendicular bisector of $AC$ at $L$. Tangent of $\Gamma$ at $C$ intersects the perpendicular bisector of $AB$ at $M$. Prove that $AK$ and $LM$ are perpendicular. by Michael Sarantis, Greece

Let $ABC$ be a triangle and let $P$ be a point on $BC$. Points $M$ and $N$ lie on $AB$ and $AC$, respectively such that $MN$ is not parallel to $BC$ and $AMP N$ is a parallelogram. Line $MN$ meets the circumcircle of $ABC$ at $R$ and $S$. Prove that the circumcircle of triangle $RP S$ is tangent to $BC$.

Let $ABCD$ be a convex plane quadrilateral and let $A_1$ denote the circumcenter of $\triangle BCD$. Define $B_1, C_1,D_1$ in a corresponding way. (a) Prove that either all of $A_1,B_1, C_1,D_1$ coincide in one point, or they are all distinct. Assuming the latter case, show that $A_1$, C1 are on opposite sides of the line $B_1D_1$, and similarly,$ B_1,D_1$ are on opposite sides of the line $A_1C_1$. (This establishes the convexity of the quadrilateral $A_1B_1C_1D_1$.) (b) Denote by $A_2$ the circumcenter of $B_1C_1D_1$, and define $B_2, C_2,D_2$ in an analogous way. Show that the quadrilateral $A_2B_2C_2D_2$ is similar to the quadrilateral $ABCD.$

$ P$ is the intersection point of diagonals of cyclic $ ABCD$. The circumcenters of $ \triangle APB$ and $ \triangle CPD$ lie on circumcircle of $ ABCD$. If $ AC \plus{} BD \equal{} 18$, then area of $ ABCD$ is ? $\textbf{(A)}\ 36 \qquad\textbf{(B)}\ \frac {81}{2} \qquad\textbf{(C)}\ \frac {36\sqrt 3}{2} \qquad\textbf{(D)}\ \frac {81\sqrt 3}{4} \qquad\textbf{(E)}\ \text{None}$

In the quadrilateral $ABCD$ is $AD \parallel BC$ and the angles $\angle A$ and $\angle D$ are acute. The diagonals intersect in $P$. The circumscribed circles of $\vartriangle ABP$ and $\vartriangle CDP$ intersect the line $AD$ again at $S$ and $T$ respectively. Call $M$ the midpoint of $[ST]$. Prove that $\vartriangle BCM$ is isosceles. [img]https://1.bp.blogspot.com/-C5MqC0RTqwY/Xy1fAavi_aI/AAAAAAAAMSM/2MXMlwb13McCYTrOHm1ZzWc0nkaR1J6zQCLcBGAsYHQ/s0/flanders%2B2016%2Bp1.png[/img]

A point $ D$ is taken on the side $ AB$ of a triangle $ ABC$. Two circles passing through $ D$ and touching $ AC$ and $ BC$ at $ A$ and $ B$ respectively intersect again at point $ E$. Let $ F$ be the point symmetric to $ C$ with respect to the perpendicular bisector of $ AB$. Prove that the points $ D,E,F$ lie on a line.

Let $A_1A_2 \ldots A_n$ be a convex polygon. Point $P$ inside this polygon is chosen so that its projections $P_1, \ldots , P_n$ onto lines $A_1A_2, \ldots , A_nA_1$ respectively lie on the sides of the polygon. Prove that for arbitrary points $X_1, \ldots , X_n$ on sides $A_1A_2, \ldots , A_nA_1$ respectively, \[\max \left\{ \frac{X_1X_2}{P_1P_2}, \ldots, \frac{X_nX_1}{P_nP_1} \right\} \geq 1.\] [i]Proposed by Nairi Sedrakyan, Armenia[/i]

Let $O$ be the circumcircle of acute triangle $ABC$, $AD$-altitude of $ABC$ ($ D \in BC$), $ AD \cap CO =E$, $M$-midpoint of $AE$, $F$-feet of perpendicular from $C$ to $AO$. Proved that point of intersection $OM$ and $BC$ lies on circumcircle of triangle $BOF$

A convex $2004$-sided polygon $P$ is given such that no four vertices are cyclic. We call a triangle whose vertices are vertices of $P$ thick if all other $2001$ vertices of $P$ lie inside the circumcircle of the triangle, and thin if they all lie outside its circumcircle. Prove that the number of thick triangles is equal to the number of thin triangles.

Triangle $ABC$ is given. Circle $ \omega $ passes through $B$, touch $AC$ in $D$ and intersect sides $AB$ and $BC$ at $P$ and $Q$ respectively. Line $PQ$ intersect $BD$ and $AC$ at $M$ and $N$ respectively. Prove that $ \omega $, circumcircle of $DMN$ and circle, touching $PQ$ in $M$ and passes through B, intersects in one point.

On the sides of a triangle $XYZ$ to the outside construct similar triangles $YDZ, EXZ ,YXF$ with circumcenters $K, L ,M$ respectively. Here are $\angle ZDY = \angle ZXE = \angle FXY$ and $\angle YZD = \angle EZX = \angle YFX$. Show that the triangle $KLM$ is similar to the triangles . [img]https://cdn.artofproblemsolving.com/attachments/e/f/fe0d0d941015d32007b6c00b76b253e9b45ca5.png[/img]

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$. Prove that there exists a point $J$ such that for any point $X$ inside $ABC$ if $AX,BX,CX$ intersect $\omega$ in $A_1,B_1,C_1$ and $A_2,B_2,C_2$ be reflections of $A_1,B_1,C_1$ in midpoints of $BC,AC,AB$ respectively then $A_2,B_2,C_2,J$ lie on a circle.

Let $ABC$ be a triangle with $\angle BAC \neq 90^{\circ}.$ Let $O$ be the circumcenter of the triangle $ABC$ and $\Gamma$ be the circumcircle of the triangle $BOC.$ Suppose that $\Gamma$ intersects the line segment $AB$ at $P$ different from $B$, and the line segment $AC$ at $Q$ different from $C.$ Let $ON$ be the diameter of the circle $\Gamma.$ Prove that the quadrilateral $APNQ$ is a parallelogram.

Let $ABCDE$ be a pentagon such that $\angle DCB < 90^{\circ} < \angle EDC$. The circle with diameter $BD$ intersects the line $BC$ again at $F$, and the circle with diameter $DE$ intersects the line $CE$ again at $G$. Prove that the second intersection ($\neq D$) of the circumcircle of $\triangle DFG$ and the circle with diameter $AD$ lies on $AC$. Proposed by [i]Petar Filipovski[/i]

In a triangle $ABC$, the bisector of angle $BAC$ intersects $BC$ at $D$. The circle $\Gamma$ through $A$ which is tangent to $BC$ at $D$ meets $AC$ again at $M$. Line $BM$ meets $\Gamma$ again at $P$. Prove that line $AP$ is a median of $\triangle ABD.$

In $\triangle ABC$, $AB>AC$, $l$ is a tangent line of the circumscribed circle of $\triangle ABC$, passing through $A$. The circle, centered at $A$ with radius $AC$, intersects $AB$ at $D$, and line $l$ at $E, F$. Prove that lines $DE, DF$ pass through the incenter and an excenter of $\triangle ABC$ respectively.

The diagonals of a cyclic quadrilateral $ABCD$ meet in a point $N.$ The circumcircles of triangles $ANB$ and $CND$ intersect the sidelines $BC$ and $AD$ for the second time in points $A_1,B_1,C_1,D_1.$ Prove that the quadrilateral $A_1B_1C_1D_1$ is inscribed in a circle centered at $N.$

Let $ABC$ be an acute-angled triangle, with $AC \neq BC$ and let $O$ be its circumcenter. Let $P$ and $Q$ be points such that $BOAP$ and $COPQ$ are parallelograms. Show that $Q$ is the orthocenter of $ABC$.

Let $\gamma$ be the incircle of $\triangle ABC$ (i.e. the circle inscribed in $\triangle ABC$) for which $AB+AC=3BC$. Let the point where $AC$ is tangent to $\gamma$ be $D$. Let the incenter of $I$. Let the intersection of the circumcircle of $\triangle BCI$ with $\gamma$ that is closer to $B$ be $P$. Show that $PID$ is collinear.

here is the most difficult and the most beautiful problem occurs in 21th iranian (2003) olympiad assume that P is n-gon ,lying on the plane ,we name its edge 1,2,..,n. if S=s1,s2,s3,.... be a finite or infinite sequence such that for each i, si is in {1,2,...,n}, we move P on the plane according to the S in this form: at first we reflect P through the s1 ( s1 means the edge which iys number is s1)then through s2 and so on like the figure below. a)show that there exist the infinite sequence S sucth that if we move P according to S we cover all the plane b)prove that the sequence in a) isn't periodic. c)assume that P is regular pentagon ,which the radius of its circumcircle is 1,and D is circle ,with radius 1.00001 ,arbitrarily in the plane .does exist a sequence S such that we move P according to S then P reside in D completely?

In a regular $2n$-gonal prism, bases $A_1A_2\cdots A_{2n}$ and $B_1B_2\cdots B_{2n}$ have circumradii equal to $R$. If the length of the lateral edge $A_1B_1$ varies, the angle between the line $A_1B_{n+1}$ and the plane $A_1A_3B_{n+2}$ is maximal for $A_1B_1=2R\cos\frac\pi{2n}$.

A triangle ABC is inscribed in a circle with center $ O$ and radius $ R$. If the inradii of the triangles $ OBC, OCA, OAB$ are $ r_{1}, r_{2}, r_{3}$ , respectively, prove that $ \frac{1}{r_{1}}+\frac{1}{r_{2}}+\frac{1}{r_{3}}\geq\frac{4\sqrt{3}+6}{R}.$