### Danube 2005-18 (Romania) 21p

geometry problems from Danube Competitions in Mathematics (a Romanian Contest)
with aops links in the names
Junior Section started in 2012
at first it was called the Danube Mathematical Cup

2005 - 2018

Senior Section

Let $\mathcal{C}$ be a circle with center $O$, and let $A$ be a point outside the circle. Let the two tangents from the point $A$ to the circle $\mathcal{C}$ meet this circle at the points $S$ and $T$, respectively. Given a point $M$ on the circle $\mathcal{C}$ which is different from the points $S$ and $T$, let the line $MA$ meet the perpendicular from the point $S$ to the line  $MO$ at $P$. Prove that the reflection of the point $S$ in the point $P$ lies on the line $MT$.

it did not take place in 2006

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 $DE \perp EF$.