Let $ABC$ be a triangle, and let $C^{\prime}$ and $A^{\prime}$ be the feet of its altitudes issuing from the vertices $C$ and $A$, respectively. Denote by $P$ the midpoint of the segment $C^{\prime}A^{\prime}$. The circumcircles of triangles $AC^{\prime}P$ and $CA^{\prime}P$ have a common point apart from $P$; denote this common point by $Q$. Prove that: [b](a)[/b] The point $Q$ lies on the circumcircle of the triangle $ABC$. [b](b)[/b] The line $PQ$ passes through the point $B$. [b](c)[/b] We have $\frac{AQ}{CQ}=\frac{AB}{CB}$. Darij

Hexagon $ABCDEF$ has a circumscribed circle and an inscribed circle. If $AB = 9$, $BC = 6$, $CD = 2$, and $EF = 4$. Find $\{DE, FA\}$.

In triangle $[ABC]$, the bisector in $C$ and the altitude passing through $B$ intersect at point $D$. Point $E$ is the symmetric of point $D$ wrt $BC$ and lies on the circle circumscribed to the triangle $[ABC]$. Prove that the triangle is $[ABC]$ isosceles.

Let $\Gamma$ be the circumcircle of an acute triangle $ABC.$ The perpendicular to $AB$ from $C$ meets $AB$ at $D$ and $\Gamma$ again at $E.$ The bisector of angle $C$ meets $AB$ at $F$ and $\Gamma$ again at $G.$ The line $GD$ meets $\Gamma$ again at $H$ and the line $HF$ meets $\Gamma$ again at $I.$ Prove that $AI = EB.$

$BB_{1}$ is a bisector of an acute triangle $ABC$. A perpendicular from $B_{1}$ to $BC$ meets a smaller arc $BC$ of a circumcircle of $ABC$ in a point $K$. A perpendicular from $B$ to $AK$ meets $AC$ in a point $L$. $BB_{1}$ meets arc $AC$ in $T$. Prove that $K$, $L$, $T$ are collinear. [i]V. Astakhov[/i]

Let $ABC$ be an acute triangle with $\angle BAC=30^{\circ}$. The internal and external angle bisectors of $\angle ABC$ meet the line $AC$ at $B_1$ and $B_2$, respectively, and the internal and external angle bisectors of $\angle ACB$ meet the line $AB$ at $C_1$ and $C_2$, respectively. Suppose that the circles with diameters $B_1B_2$ and $C_1C_2$ meet inside the triangle $ABC$ at point $P$. Prove that $\angle BPC=90^{\circ}$ .

In a triangle $ABC$, $\angle BAC =90^{\circ}$. Point $D$ lies on the side $BC$ and satisfies $\angle BDA=2\angle BAD$. Prove that \[\frac{2}{AD}=\frac{1}{BD}+\frac{1}{CD} \]

Let $ABC$ be a triangle with $AB=26$, $AC=28$, $BC=30$. Let $X$, $Y$, $Z$ be the midpoints of arcs $BC$, $CA$, $AB$ (not containing the opposite vertices) respectively on the circumcircle of $ABC$. Let $P$ be the midpoint of arc $BC$ containing point $A$. Suppose lines $BP$ and $XZ$ meet at $M$ , while lines $CP$ and $XY$ meet at $N$. Find the square of the distance from $X$ to $MN$. [i]Proposed by Michael Kural[/i]

For an acute triangle $ABC$, let $O$ be its circumcenter, and let $O_A,O_B,O_C$ be the circumcenter of $BCO,CAO,ABO$ respectively. Show that $AO_A,BO_B,CO_C$ are concurrent.

Given a pyramid whose base is an $n$-gon inscribable in a circle, let $H$ be the projection of the top vertex of the pyramid to its base. Prove that the projections of $H$ to the lateral edges of the pyramid lie on a circle.

Let $ ABC$ be an acute angled triangle with $ \angle B > \angle C$. Let $ D$ be the foot of the altitude from $ A$ on $ BC$, and let $ E$ be the foot of the perpendicular from $ D$ on $ AC$. Let $ F$ be a point on the segment $ (DE)$. Show that the lines $ AF$ and $ BF$ are perpendicular if and only if $ EF\cdot DC \equal{} BD \cdot DE$.

Let $ ABC $ be an acute triangle. Denote by $ D $ the foot of the perpendicular line drawn from the point $ A $ to the side $ BC $, by $M$ the midpoint of $ BC $, and by $ H $ the orthocenter of $ ABC $. Let $ E $ be the point of intersection of the circumcircle $ \Gamma $ of the triangle $ ABC $ and the half line $ MH $, and $ F $ be the point of intersection (other than $E$) of the line $ ED $ and the circle $ \Gamma $. Prove that $ \tfrac{BF}{CF} = \tfrac{AB}{AC} $ must hold. (Here we denote $XY$ the length of the line segment $XY$.)

$I$ is the incenter of triangle $ABC$. perpendicular from $I$ to $AI$ meet $AB$ and $AC$ at ${B}'$ and ${C}'$ respectively . Suppose that ${B}''$ and ${C}''$ are points on half-line $BC$ and $CB$ such that $B{B}''=BA$ and $C{C}''=CA$. Suppose that the second intersection of circumcircles of $A{B}'{B}''$ and $A{C}'{C}''$ is $T$. Prove that the circumcenter of $AIT$ is on the $BC$.

Let $AB=AC$ in $\triangle ABC$, and let $D$ be a point on segment $AB$. The tangent at $D$ to the circumcircle $\omega$ of $BCD$ hits $AC$ at $E$. The other tangent from $E$ to $\omega$ touches it at $F$, and $G=BF \cap CD$, $H=AG \cap BC$. Prove that $BH=2HC$. [i]Proposed by David Stoner[/i]

Let $ABC$ be an acute triangle with $\displaystyle{AB<AC<BC}$ inscribed in circle $ \displaystyle{c(O,R)}$.The excircle $\displaystyle{(c_A)}$ has center $\displaystyle{I}$ and touches the sides $\displaystyle{BC,AC,AB}$ of the triangle $ABC$ at $\displaystyle{D,E,Z} $ respectively.$ \displaystyle{AI}$ cuts $\displaystyle{(c)}$ at point $M$ and the circumcircle $\displaystyle{(c_1)}$ of triangle $\displaystyle{AZE}$ cuts $\displaystyle{(c)}$ at $K$.The circumcircle $\displaystyle{(c_2)}$ of the triangle $\displaystyle{OKM}$ cuts $\displaystyle{(c_1)} $ at point $N$.Prove that the point of intersection of the lines $AN,KI$ lies on $ \displaystyle{(c)}$.

Let $ABCD$ be a parallelogram with at least an angle not equal to $90^o$ and $k$ the circumcircle of the triangle $ABC$. Let $E$ be the diametrically opposite point of $B$. Show that the circumcircle of the triangle $ADE$ and $k$ have the same radius.

The perpendicular bisectors of the sides AB and BC of a triangle ABC meet the lines BC and AB at the points X and Z, respectively. The angle bisectors of the angles XAC and ZCA intersect at a point B'. Similarly, define two points C' and A'. Prove that the points A', B', C' lie on one line through the incenter I of triangle ABC. [i]Extension:[/i] Prove that the points A', B', C' lie on the line OI, where O is the circumcenter and I is the incenter of triangle ABC. Darij

According to the estimated number of participants who gave a correct solution, this was the hardest (!) problem from today's paper. So here is this great German killer - be warned! Given a circle k and three pairwisely distinct points A, B, C on this circle. Let h and g be the perpendiculars to the line BC at the points B and C. The perpendicular bisector of the segment AB meets the line h at a point F; the perpendicular bisector of the segment AC meets the line g at a point G. Prove that the product $BF\cdot CG$ is independent from the position of the point A, as long as the points B and C stay fixed. The actual problem behind the problem: Why on hell should the points B and C stay fixed? Darij

Let ${\cal C}_1$ and ${\cal C}_2$ be concentric circles, with ${\cal C}_2$ in the interior of ${\cal C}_1$. From a point $A$ on ${\cal C}_1$ one draws the tangent $AB$ to ${\cal C}_2$ ($B\in {\cal C}_2$). Let $C$ be the second point of intersection of $AB$ and ${\cal C}_1$, and let $D$ be the midpoint of $AB$. A line passing through $A$ intersects ${\cal C}_2$ at $E$ and $F$ in such a way that the perpendicular bisectors of $DE$ and $CF$ intersect at a point $M$ on $AB$. Find, with proof, the ratio $AM/MC$.

Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$. [i]Proposed by Hojoo Lee, Korea[/i]

Consider an isosceles triangle $ABC$ with sides $BC = 30, CA = AB = 20$. Let $D$ be the foot of the perpendicular from $A$ to $BC$, and let $M$ be the midpoint of $AD$. Let $PQ$ be a chord of the circumcircle of triangle $ABC$, such that $M$ lies on $PQ$ and $PQ$ is parallel to $BC$. The length of $PQ$ is:

Given a cyclic quadrilateral $ABCD$ with the circumcenter $O$, with $BC$ and $AD$ not parallel. Let $P$ be the intersection of $AC$ and $BD$. Let $E$ be the intersection of the rays $AB$ and $DC$. Let $I$ be the incenter of $EBC$ and the incircle of $EBC$ touches $BC$ at $T_1$. Let $J$ be the excenter of $EAD$ that touches $AD$ and the excircle of $EAD$ that touches $AD$ touches $AD$ at $T_2$. Let $Q$ be the intersection between $IT_1$ and $JT_2$. Prove that $O,P,Q$ are collinear.

Let $ABC$ be an acute scalene triangle with circumcircle $\omega$ and incenter $I$. Suppose the orthocenter $H$ of $BIC$ lies inside $\omega$. Let $M$ be the midpoint of the longer arc $BC$ of $\omega$. Let $N$ be the midpoint of the shorter arc $AM$ of $\omega$. Prove that there exists a circle tangent to $\omega$ at $N$ and tangent to the circumcircles of $BHI$ and $CHI$.

Let $ \triangle ABC$ be a triangle with incircle $ \omega(I,r)$and circumcircle $ \zeta(O,R)$.Let $ l_{a}$ be the angle bisector of $ \angle BAC$.Denote $ P\equal{}l_{a}\cap\zeta$.Let $ D$ be the point of tangency $ \omega$ with $ [BC]$.Denote $ Q\equal{}PD\cap\zeta$.Show that $ PI\equal{}QI$ if $ PD\equal{}r$.

Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\Omega$. Let the lines $AB$ and $CD$ intersect at $P$, and the lines $AD$ and $BC$ intersect at $Q$. Let then the circumcircle of the triangle $\triangle APQ$ intersect $\Omega$ at $R \neq A$. Prove that the line $CR$ goes through the midpoint of the segment $PQ$.