### Danube 2007-18 (Romania) 20p

geometry problems from Danube Competitions in Mathematics (a Romanian Contest)
with aops links in the names
Junior Section started in 2012

2007 - 2018

Senior Section
Let $ABCD$ be an inscribed quadrilateral and let $E$ be the midpoint of the diagonal $BD$. Let $\Gamma_1,\Gamma_2,\Gamma_3,\Gamma_4$ be the circumcircles of triangles $AEB$, $BEC$, $CED$ and $DEA$ respectively. Prove that if $\Gamma_4$ is tangent to the line $CD$, then $\Gamma_1,\Gamma_2,\Gamma_3$ are tangent to the lines $BC,AB,AD$ respectively.

2008 Danube P2
In a triangle $ABC$  let  $A_1$ be the midpoint of side $BC$. Draw circles with centers $A, A1$ and radii $AA_1, BC$   respectively  and let $A'A''$ be their common chord. Similarly denote the segments $B'B''$ and $C'C''$. Show that lines $A'A'', B'B'''$ and $C'C''$ are concurrent.

Let be $\triangle ABC$ .Let $A'$, $B'$, $C'$ be the foot of perpendiculars from $A$, $B$ and $C$ respectively. The points $E$ and $F$ are on the sides $CB'$ and $BC'$ respectively, such that $B'E\cdot C'F = BF\cdot CE$. Show that $AEA'F$ is cyclic.

Let $\triangle ABC$ and denote by $G$ the intersection of it's medians.Let $A_{1}$,$B_{1}$,$C_{1}$ respectively, the foot of perpendiculars from $G$ to $BC$ , $AC$, $AB$ respectively. Denote by $A_{2}$ ,$B_{2}$ and $C_{2}$ respectively the symmetric points of  $A_{1}$,$B_{1}$,$C_{1}$ respectively, with respect to point $G$.Prove that $AA_{2}$,$BB_{2}$ and $CC_{2}$ are concurrent.

2011 Danube P1
Let $ABCM$ be a quadrilateral and $D$ be an interior point such that $ABCD$ is a parallelogram. It is known that $\angle AMB =\angle CMD$. Prove that $\angle MAD =\angle MCD$.

2012 Danube Senior P2
Let ABC be an acute triangle and let $A_1$, $B_1$, $C_1$ be points on the sides $BC, CA$ and $AB$, respectively. Show that the triangles $ABC$ and $A_1B_1C_1$ are similar ($\angle A = \angle A_1, \angle B = \angle B_1,\angle C = \angle C_1$) if and only if the orthocentre of the triangle $A_1B_1C_1$ and the circumcentre of the triangle $ABC$ coincide.

2013 Danube Senior P1
Given six points on a circle, $A, a, B, b, C, c$, show that the Pascal lines of the hexagrams $AaBbCc, AbBcCa, AcBaCb$ are concurrent.

2014 Danube Senior P1
Two circles $\gamma_1$ and $\gamma_2$ cross one another at two points; let $A$ be one of these points. The tangent to $\gamma_1$ at $A$ meets again $\gamma_2$ at $B$, the tangent to $\gamma_2$ at $A$ meets again $\gamma_1$ at $C$, and the line $BC$ meets again $\gamma_1$ and $\gamma_2$ at $D_1$ and $D_2$, respectively. Let $E_1$ and $E_2$ be interior points of the segments $AD_1$ and $AD_2$, respectively, such that $AE_1 = AE_2$. The lines $BE_1$ and $AC$ meet at $M$, the lines $CE_2$ and $AB$ meet at $N$, and the lines $MN$ and $BC$ meet at $P$. Show that the line $PA$ is tangent to the circle $ABC$.

Let $ABCD$ be a cyclic quadrangle, let the diagonals $AC$ and $BD$ cross at $O$, and let $I$ and $J$ be the incentres of the triangles $ABC$ and $ABD$, respectively. The line $IJ$ crosses the segments $OA$ and $OB$ at $M$ and $N$, respectively. Prove that the triangle $OMN$ is isosceles.

Let $ABC$ be a triangle, $D$ the foot of the altitude from $A$ and $M$ the midpoint of the side $BC$. Let $S$ be a point on the closed segment $DM$ and let $P, Q$ the projections of $S$ on the lines $AB$ and $AC$ respectively. Prove that the length of the segment $PQ$ does not exceed one quarter the perimeter of the triangle $ABC$.

Let $O,H$ be the circumcenter and the orthocenter of triangle $ABC$. Let $F$ be the foot of the perpendicular from C onto AB, and $M$ the midpoint of $CH$. Let N be the foot of the perpendicular from C onto the parallel through H at $OM$. Let $D$ be on $AB$ such that $CA=CD$. Let $BN$ intersect $CD$ at $P$. Let $PH$ intersect $CA$ at $Q$. Prove that $QF\perp OF$.

Let $ABC$ be an acute non isosceles triangle. The angle bisector of angle $A$ meets again the circumcircle of the triangle $ABC$ in $D$. Let $O$ be the circumcenter of the triangle $ABC$. The angle bisectors of $\angle AOB$, and $\angle AOC$ meet the circle $\gamma$ of diameter $AD$ in $P$ and $Q$ respectively. The line $PQ$ meets the perpendicular bisector of $AD$ in $R$. Prove that $AR // BC$.

Junior Section
(started in 2012)
2012 Danube Junior P3
Let $ABC$ be a triangle with $\angle BAC = 90^o$. Angle bisector of the $\angle CBA$  intersects the segment $(AB)$ at point $E$. If there exists $D \in (CE)$ so  that $\angle DAC = \angle BDE =x^o$ , calculate $x$.

2013 Danube Junior P4
Let $ABCD$ be a rectangle   with $AB \ne BC$ and  the center the point $O$. Perpendicular from $O$ on $BD$ intersects lines $AB$ and $BC$ in points $E$ and $F$ respectively. Points $M$ and $N$ are midpoints of segments $[CD]$ and  $[AD]$ respectively. Prove that $FM \perp EN$ .

2014 Danube Junior P3
Let $ABC$ be a triangle with $\angle A<90^o, AB \ne AC$. Denote  $H$ the orthocenter of triangle $ABC$, $N$ the midpoint of segment $[AH]$, $M$ the midpoint of segment $[BC]$ and $D$ the intersection point of the angle bisector of $\angle BAC$ with the segment $[MN]$. Prove that $<ADH=90^o$

2015 Danube Junior P4
Let $ABCD$ be a rectangle  with $AB\ge BC$  Point $M$ is located on the side $(AD)$, and the perpendicular bisector of  $[MC]$ intersects the line $BC$ at the point $N$. Let ${Q} =MN\cup AB$ . Knowing that $\angle MQA= 2\cdot \angle BCQ$, show that the quadrilateral $ABCD$ is a square.

Let $ABC$ be a triangle with $AB < AC$,$I$ its incenter, and $M$ the midpoint of the side $BC$. If $IA = IM$, determine the smallest possible value of the angle $AIM$.

Given an acute triangle $ABC$ with orthocenter $H$, let $A_1,B_1,C_1$ be the feet of the altitudes drawn from $A,B,C$, respectively. Denote by $P$ the intersection of $AB$ with the perpendicular from $H$ onto $A_1C_1$, and by $Q$ the intersection of $AC$ with the perpendicular from $H$ onto $A_1B_1$. Prove that the midpoint of segment $PQ$ lies on the perpendicular from $H$ onto $B_1C_1$.

Let $ABC$ be a triangle such that in its interior there exists a point $D$ with $\angle DAC = \angle DCA = 30^o$ and $\angle DBA = 60^o$. Denote $E$ the midpoint of the segment $BC$, and take $F$ on the segment $AC$ so that $AF = 2FC$. Prove that $DE \perp EF$.