### Stars Of Mathematics 2007-18 (Romania) 18p

geometry problems from Stars Of Mathematics (Romanian)
Stelele Matematici , ICHB Bucuresti

Senior part started in 2007, junior part in 2011

junior 2011-2018

Let $ABC$ be an acute-angled, not equilateral triangle, where vertex $A$ lies on the perpendicular bisector of the segment $HO$, joining the orthocentre $H$ to the circumcentre $O$. Determine all possible values for the measure of angle $A$.
(U.S.A. - 1989 IMO Shortlist)
2012 Stars of Mathematics Juniors P1
Let $\ell$ be a line in the plane, and a point $A \not \in \ell$. Determine the locus of the points $Q$ in the plane, for which there exists a point $P\in \ell$ so that $AQ=PQ$ and $\angle PAQ = 45^{\circ}$.

(Dan Schwarz)
2013 Stars of Mathematics Juniors P2
Three points inside a rectangle determine a triangle. A fourth point is taken inside the triangle. Prove that at least one of the three concave quadrilaterals formed by these four points has perimeter lesser than that of the rectangle.
(Dan Schwarz)
2015 Stars of Mathematics Juniors P3
Let $ABCD$ be cyclic quadrilateral,let $\gamma$ be it's circumscribed circle and let $M$ be the midpoint of arc $AB$ of $\gamma$,which does not contain points $C,D$.The line that passes through $M$ and the intersection point of diagonals $AC,BD$,intersects $\gamma$ in $N\neq M$.
Let $P,Q$ be two points situated on $CD$,such that $\angle{AQD}=\angle{DAP}$ and $\angle{BPC}=\angle{CBQ}$.Prove that circles $\odot(NPQ)$ and $\gamma$ are tangent.

(Flavian Georgescu)
2016 Stars of Mathematics Juniors P4
Let $ABC$ be an acute angled triangle with $AB <AC$, $I$ be the center of its incircle, $D, E, F$ touch points of the incircle with the sides $BC, CA$ and $AB$ respectively,  $X$ the midpoint of the arc  $BAC$ of the circumcircle of the triangle $ABC$. $P$ is the projection of $D$ on $EF, Q$ is the projection of $A$ on $ID$.
a) Prove that that the lines $IX$ and $PQ$ are parallel.
b) If $Y$ is the second point of intersection between the circle of diameter $AI$ and the circumcircle of  the triangle $ABC$,  prove that the lines $XQ$ and $PI$ intersect at $Y$.

Let $ABC$ be an acute triangle in which $AB < AC$. Let $M$ be the midpoint of the side $BC$ and consider $D$ an arbitrary point of the line segment $AM$. Let $E$ be a point of the line segment $BD$ and consider the point $F$ of the line $AB$ such that lines $EF$ and $BC$ are parallel. If the orthocenter, $H$, of the triangle $ABC$ lies at the intersection point of lines $AE$ and $DF$, prove that the angle bisectors of $\angle BAC$ and $\angle BDC$ meet on the line $BC$.

2018 Stars of Mathematics Juniors P3
Let be an isosceles trapezoid such that its smaller base is equal to its legs, and a rhombus that has each of its vertexes on a different side of the trapezoid. Prove that the smaller angles of the trapezoid are equal to the smaller ones of the rhombus.

senior 2007-2017

2007 Stars of Mathematics Seniors P3
Let $ABC$ be a triangle and $A_1,B_1,C_1$ be the feet of the altitudes from $A,B,C$. Let $A_2$ respectively $A_3$, be the orthogonal projection of $A_1$ onto $AB$, respectively $AC$, points $B_2,B_3$ and $C_2,C_3$ are defined in an analogous way. The lines $B_2B_3$ and $C_2C_3$ meet at $A_4$, the  lines $C_2C_3$ and $A_2A_3$ meet at $B_4$, while  the lines $A_2A_3$ and $B_2B_3$ meet at $C_4$. Show that the lines $AA_4,BB_4$ and $CC_4$ are concurrent.

2007 Stars of Mathematics Seniors P7
Let $A_0A_1...A_{n-1}$ be a regular $n$-gon. For each index $i$, consider a point $B_i$ lying on the side $A_iA_{i+1}$, such that $A_iB_i<A_iA_{i+1}/2$, and a point $C_i$ lying on the segment $A_iB_i$ (indices are reduced modulo $n$). Show that the perimeter of the polygon $C_0C_1...C_{n-1}$ is at least as large as the perimeter of the polygon $B_0B_1...B_{n-1}$.

2008 Stars of Mathematics Seniors P3
Consider a convex quadrilateral, and the incircles of the triangles determined by one of its diagonals. Prove that the tangency points of the incircles with the diagonal are symmetrical with respect to the midpoint of the diagonal if and only if the line of the incenters passes through the crossing point of the diagonals.
(Dan Schwarz)
2009 Stars of Mathematics Seniors P2
Let $\omega$ be a circle in the plane and $A,B$ two points lying on it. We denote by $M$ the midpoint of $AB$ and let $P \ne M$ be a new point on $AB$. Build circles $\gamma$ and $\delta$ tangent to $AB$ at $P$ and to $\omega$ at $C$, respectively $D$. Consider $E$ to be the point diametrically opposed to $D$ in $\omega$. Prove that the circumcenter of $\triangle BMC$ lies on the line $BE$.

2010 Stars of Mathematics Seniors P2
Let $ABCD$ be a square and let the points $M$ on $BC$, $N$ on $CD$, $P$ on $DA$, be such that $\angle (AB,AM)=x,\angle (BC,MN)=2x,\angle (CD,NP)=3x$.
1) Show that for any $0\le x\le 22.5$, such a configuration uniquely exists, and that $P$ ranges over the whole segment $DA$;
2) Determine the number of angles $0\le x\le 22.5$ for which$\angle (DA,PB)=4x$.

(Dan Schwarz)
Let $ABC$ be an acute-angled triangle with $AB \neq BC$, $M$ the midpoint of $AC$, $N$ the point where the median $BM$ meets again the circumcircle of $\triangle ABC$, $H$ the orthocentre of $\triangle ABC$, $D$ the point on the circumcircle for which $\angle BDH = 90^{\circ}$, and $K$ the point that makes $ANCK$ a parallelogram. Prove the lines $AC$, $KH$, $BD$ are concurrent.
(I. Nagel)
2012 Stars of Mathematics Seniors P2
Let $\ell$ be a line in the plane, and a point $A \not \in \ell$. Also let $\alpha \in (0, \pi/2)$ be fixed. Determine the locus of the points $Q$ in the plane, for which there exists a point $P\in \ell$ such that $AQ=PQ$ and $\angle PAQ = \alpha$.
(Dan Schwarz)
2013 Stars of Mathematics Seniors P2
Three points inside a rectangle determine a triangle. A fourth point is taken inside the triangle.
i) Prove at least one of the three concave quadrilaterals formed by these four points has perimeter lesser than that of the rectangle.
ii) Assuming the three points inside the rectangle are three corners of it, prove at least two of the three concave quadrilaterals formed by these four points have perimeters lesser than that of the rectangle.
(Dan Schwarz)
Let $\gamma,\gamma_0,\gamma_1,\gamma_2$ be four circles in plane,such that $\gamma_i$ is interiorly tangent to $\gamma$ in point $A_i$,and $\gamma_i$ and $\gamma_{i+1}$ are exteriorly tangent in point $B_{i+2}$,$i=0,1,2$(the indexes are reduced modulo $3$).The tangent in $B_i$,common for circles $\gamma_{i-1}$ and $\gamma_{i+1}$,intersects circle $\gamma$ in point $C_i$,situated in the opposite semiplane of $A_i$ with respect to line $A_{i-1}A_{i+1}$.Prove that the three lines $A_iC_i$ are concurrent.

2016 Stars of Mathematics Seniors P3
Let $ABC$ be a triangle, let $M_A$ be the midpoint of the side $BC$, and let $P_A$ be the orthogonal projection of $A$ on the line $BC$, similarly, define $M_B, P_B$ and $M_C, P_C$. The lines $M_BM_C$ and $P_BP_C$ meet at $S_A$, and the tangent of the circle $ABC$ at $A$ meets the line $BC$ at T$_A$, similarly, define $S_B, T_B$ and $S_C, T_C$. Show that the perpendiculars through $A, B, C$ to the lines $S_AT_A, S_BT_B, S_CT_C$, respectively, are concurrent.
(Flavian Georgescu)
2017 Stars of Mathematics Seniors P2
Let $ABC$ be a triangle and $O$ be its circumcenter. Let $P,Q$ be two points in the interior of $ABC$, $R$ the reflection of $O$ wrt of the midpoint of $PQ$, and $S$ the reflection of $R$ wrt of the Euler circle center. Let $\omega$ be the circle that passes through $P,Q$ such that $\omega$ and the circumcenter are orthogonal. Let $(OP,(OQ$ intersect $\omega$ in $P,P’$ and $Q,Q’$. Let $P’Q,PQ’$ intersect at $T$. Prove that if $P,Q$ are isogonal conjugates, then $S,T$ are also isogonal conjugates.