Bosnia & Herzegovina Regional 2008-18 40p

geometry problems from Regional Olympiads of  Bosnia and Herzegovina
with aops links

2008 - 2018

2008 Bosnia and Herzegovina Regional 9.1
Squares $BCA_{1}A_{2}$ , $CAB_{1}B_{2}$ , $ABC_{1}C_{2}$ are outwardly drawn on sides of triangle $\triangle ABC$. If $AB_{1}A'C_{2}$ , $BC_{1}B'A_{2}$ , $CA_{1}C'B_{2}$ are parallelograms then prove that:
(i) Lines $BC$ and $AA'$ are orthogonal.
(ii)Triangles $\triangle ABC$ and $\triangle A'B'C'$ have common centroid

2008 Bosnia and Herzegovina Regional 10.1
Given is an acute angled triangle $\triangle ABC$ with side lengths $a$, $b$ and $c$ (in an usual way) and circumcenter $O$. Angle bisector of angle $\angle BAC$ intersects circumcircle at points $A$ and $A_{1}$. Let $D$ be projection of point $A_{1}$ onto line $AB$, $L$ and $M$ be midpoints of $AC$ and $AB$ , respectively.
(i) Prove that $AD=\frac{1}{2}(b+c)$
(ii) If triangle $\triangle ABC$ is an acute angled prove that $A_{1}D=OM+OL$

2008 Bosnia and Herzegovina Regional 11.1
Two circles with centers $S_{1}$ and $S_{2}$ are externally tangent at point $K$. These circles are also internally tangent to circle $S$ at points $A_{1}$ and $A_{2}$, respectively. Denote by $P$one of the intersection points of $S$ and common tangent to $S_{1}$ and $S_{2}$ at $K$.Line $PA_{1}$ intersects $S_{1}$ at $B_{1}$ while $PA_{2}$ intersects $S_{2}$ at $B_{2}$.
Prove that $B_{1}B_{2}$ is common tangent of circles $S_{1}$ and $S_{2}$.

2008 Bosnia and Herzegovina Regional 12.1
Given are three pairwise externally tangent circles $K_{1}$ , $K_{2}$ and $K_{3}$. denote by $P_{1}$ tangent point of $K_{2}$ and $K_{3}$ and by $P_{2}$ tangent point of $K_{1}$ and $K_{3}$. Let $AB$ ($A$ and $B$ are different from tangency points) be a diameter of circle $K_{3}$. Line $AP_{2}$ intersects circle $K_{1}$ (for second time) at point $X$ and line $BP_{1}$ intersects circle $K_{2}$(for second time) at $Y$. If $Z$ is intersection point of lines $AP_{1}$ and $BP_{2}$ prove that points $X$, $Y$ and $Z$ are collinear.

2009 Bosnia and Herzegovina Regional 9.4
Let $C$ be a circle with center $O$ and radius $R$. From point $A$ of circle $C$ we construct a tangent $t$ on circle $C$. We construct line $d$ through point $O$ whch intersects tangent $t$ in point $M$ and circle $C$ in points $B$ and $D$ ($B$ lies between points $O$ and $M$). If $AM=R\sqrt{3}$, prove:
$a)$ Triangle $AMD$ is isosceles
$b)$ Circumcenter of $AMD$ lies on circle $C$

2009 Bosnia and Herzegovina Regional 10.1
In triangle $ABC$ such that $\angle ACB=90^{\circ}$, let point $H$ be foot of perpendicular from point $C$ to side $AB$. Show that sum of radiuses of incircles of $ABC$, $BCH$ and $ACH$ is $CH$

2009 Bosnia and Herzegovina Regional 11.1
In triangle $ABC$ holds $\angle ACB = 90^{\circ}$, $\angle BAC = 30^{\circ}$ and $BC=1$. In triangle $ABC$ is inscribed equilateral triangle (every side of a triangle $ABC$ contains one vertex of inscribed triangle). Find the least possible value of side of inscribed equilateral triangle

2009 Bosnia and Herzegovina Regional 12.2
Let $ABC$ be an equilateral triangle such that length of its altitude is $1$. Circle with center on the same side of line $AB$ as point $C$ and radius $1$ touches side $AB$. Circle rolls on the side $AB$. While the circle is rolling, it constantly intersects sides $AC$ and $BC$. Prove that length of an arc of the circle, which lies inside the triangle, is constant.

2010 Bosnia and Herzegovina Regional 9.2
In convex quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at point $O$ at angle $90^{\circ}$. Let $K$, $L$, $M$ and $N$ be orthogonal projections of point $O$ to sides $AB$, $BC$, $CD$ and $DA$ of quadrilateral $ABCD$. Prove that $KLMN$ is cyclic

2010 Bosnia and Herzegovina Regional 10.2
It is given acute triangle $ABC$ with orthocenter at point $H$. Prove that $$AH \cdot h_a+BH \cdot h_b+CH \cdot h_c=\frac{a^2+b^2+c^2}{2}$$ where $a$, $b$ and $c$ are sides of a triangle, and $h_a$, $h_b$ and $h_c$ altitudes of $ABC$

2010 Bosnia and Herzegovina Regional 11.2
Angle bisector from vertex $A$ of acute triangle $ABC$ intersects side $BC$ in point $D$, and circumcircle of $ABC$ in point $E$ (different from $A$). Let $F$ and $G$ be foots of perpendiculars from point $D$ to sides $AB$ and $AC$. Prove that area of quadrilateral $AEFG$ is equal to the area of triangle $ABC$

2010 Bosnia and Herzegovina Regional 12.4
Let $AA_1$, $BB_1$ and $CC_1$ be altitudes of triangle $ABC$ and let $A_1A_2$, $B_1B_2$ and $C_1C_2$ be diameters of Euler circle of triangle $ABC$. Prove that lines $AA_2$, $BB_2$ and $CC_2$ are concurrent

2011 Bosnia and Herzegovina Regional 9.3
Triangle $AOB$ is rotated in plane around point $O$ for $90^{\circ}$ and it maps in triangle $A_1OB_1$ ($A$ maps to $A_1$, $B$ maps to $B_1$). Prove that median of triangle $OAB_1$ of side $AB_1$ is orthogonal to $A_1B$

2011 Bosnia and Herzegovina Regional 10.3
Let $I$ be the incircle and $O$ a circumcenter of triangle $ABC$ such that $\angle ACB=30^{\circ}$. On sides $AC$ and $BC$ there are points $E$ and $D$, respectively, such that  $EA=AB=BD$. Prove that $DE=IO$ and $DE \perp IO$

2011 Bosnia and Herzegovina Regional 11.3 12.1
Let $AD$ and $BE$ be angle bisectors in triangle $ABC$. Let $x$, $y$ and $z$ be distances from point $M$, which lies on segment $DE$, from sides $BC$, $CA$ and $AB$, respectively. Prove that $z=x+y$

2012 Bosnia and Herzegovina Regional 9.4
Let $S$ be an incenter of triangle $ABC$ and let incircle touch sides $AC$ and $AB$ in points $P$ and $Q$, respectively. Lines $BS$ and $CS$ intersect line $PQ$ in points $M$ and $N$, respectively. Prove that points $M$, $N$, $B$ and $C$ are concyclic.

2012 Bosnia and Herzegovina Regional 10.3
Quadrilateral $ABCD$ is cyclic. Line through point $D$ parallel with line $BC$ intersects $CA$ in point $P$, line $AB$ in point $Q$ and circumcircle of $ABCD$ in point $R$. Line through point $D$ parallel with line $AB$ intersects $AC$ in point $S$, line $BC$ in point $T$ and circumcircle of $ABCD$ in point $U$. If $PQ=QR$, prove that $ST=TU$

2012 Bosnia and Herzegovina Regional 11.4 12.3
In triangle $ABC$ point $O$ is circumcenter. Point $T$ is centroid of $ABC$, and points $D$, $E$ and $F$ are circumcenters of triangles $TBC$, $TCA$ and $TAB$. Prove that $O$ is centroid of $DEF$

2013 Bosnia and Herzegovina Regional 9.2
In triangle $ABC$, $\angle ACB=50^{\circ}$ and $\angle CBA=70^{\circ}$. Let $D$ be a foot of perpendicular from point $A$ to side $BC$, $O$ circumcenter of $ABC$ and $E$ antipode of $A$ in circumcircle $ABC$. Find $\angle DAE$

2013 Bosnia and Herzegovina Regional 10.2
In circle with radius $10$, point $M$ is on chord $PQ$ such that $PM=5$ and $MQ=10$. Through point $M$ we draw chords $AB$ and $CD$, and points $X$ and $Y$ are intersection points of chords $AD$ and $BC$ with chord $PQ$ (see picture), respectively. If $XM=3$ find $MY$

2013 Bosnia and Herzegovina Regional 11.3 12.3
Convex quadrilateral is divided by diagonals into four triangles with congruent inscribed circles. Prove that this quadrilateral is rhombus.

2014 Bosnia and Herzegovina Regional 9.3
In triangle $ABC$ $(b \geq c)$, point $E$ is the midpoint of shorter arc $BC$. If $D$ is the point such that $ED$ is the diameter of circumcircle $ABC$, prove that $\angle DEA = \frac{1}{2}(\beta-\gamma)$

2014 Bosnia and Herzegovina Regional 10.3
Let $ABCD$ be a parallelogram. Let $M$ be a point on the side $AB$ and $N$ be a point on the side $BC$ such that the segments $AM$ and $CN$ have equal lengths and are non-zero. The lines $AN$ and $CM$ meet at $Q$.
Prove that the line $DQ$ is the bisector of the angle $\measuredangle ADC$.

Alternative formulation.
Let $ABCD$ be a parallelogram. Let $M$ and $N$ be points on the sides $AB$ and $BC$, respectively, such that $AM=CN\neq 0$. The lines  $AN$ and $CM$ intersect at a point $Q$.
Prove that the point $Q$ lies on the bisector of the angle $\measuredangle ADC$.

2014 Bosnia and Herzegovina Regional 11.3 12.3
Excircle of triangle $ABC$ to side $AB$ of triangle $ABC$ touches side $AB$ in point $D$. Determine ratio $AD : BD$ if $\angle CAB = 2 \angle ADC$

2015 Bosnia and Herzegovina Regional 9.3
In parallelogram $ABCD$ holds $AB=BD$. Let $K$ be a point on $AB$, different from $A$, such that $KD=AD$. Let $M$ be a point symmetric to $C$ with respect to $K$, and $N$ be a point symmetric to point $B$ with respect to $A$. Prove that $DM=DN$

2015 Bosnia and Herzegovina Regional 10.3
Let $ABC$ be a triangle with incenter $I$. Line $AI$ intersects circumcircle of $ABC$ in points $A$ and $D$, $(A \neq D)$. Incircle of $ABC$ touches side $BC$ in point $E$ . Line $DE$ intersects circumcircle of $ABC$ in points $D$ and $F$, $(D \neq F)$. Prove that $\angle AFI = 90^{\circ}$

2015 Bosnia and Herzegovina Regional 11.3
Let $F$ be an intersection point of altitude $CD$ and internal angle bisector $AE$ of right angled triangle $ABC$, $\angle ACB = 90^{\circ}$. Let $G$ be an intersection point of lines $ED$ and $BF$. Prove that area of quadrilateral $CEFG$ is equal to area of triangle $BDG$

2015 Bosnia and Herzegovina Regional 12.3
Let $O$ and $I$ be circumcenter and incenter of triangle $ABC$. Let incircle of $ABC$ touches sides $BC$, $CA$ and $AB$ in points $D$, $E$ and $F$, respectively. Lines $FD$ and $CA$ intersect in point $P$, and lines $DE$ and $AB$ intersect in point $Q$. Furthermore, let $M$ and $N$ be midpoints of $PE$ and $QF$. Prove that $OI \perp MN$

2016 Bosnia and Herzegovina Regional 9.2
Let $ABC$ be an isosceles triangle such that $\angle BAC = 100^{\circ}$. Let $D$ be an intersection point of angle bisector of $\angle ABC$ and side $AC$, prove that $AD+DB=BC$

2016 Bosnia and Herzegovina Regional 10.3
Let $AB$ be a diameter of semicircle $h$. On this semicircle there is point $C$, distinct from points $A$ and $B$. Foot of perpendicular from point $C$ to side $AB$ is point $D$. Circle $k$ is outside the triangle $ADC$ and at the same time touches semicircle $h$ and sides $AB$ and $CD$. Touching point of $k$ with side $AB$ is point $E$, with semicircle $h$ is point $T$ and with side $CD$ is point $S$
$a)$ Prove that points $A$, $S$ and $T$ are collinear
$b)$ Prove that $AC=AE$

2016 Bosnia and Herzegovina Regional 11.3
$h_a$,  $h_b$ and  $h_c$ are altitudes,  $t_a$,  $t_b$ and  $t_c$ are medians of acute triangle, $r$ radius of incircle, and $R$ radius of circumcircle of acute triangle $ABC$. Prove that $\frac{t_a}{h_a}+\frac{t_b}{h_b}+\frac{t_c}{h_c} \leq 1+ \frac{R}{r}$

2016 Bosnia and Herzegovina Regional 12.3
Circle of radius $R_1$ is inscribed in an acute angle $\alpha$. Second circle with radius $R_2$ touches one of the sides forming the angle $\alpha$ in same point as first circle and intersects the second side in points $A$ and $B$, such that centers of both circles lie inside angle $\alpha$. Prove that $$AB=4\cos{\frac{\alpha}{2}}\sqrt{(R_2-R_1)\left(R_1 \cos^2 \frac{\alpha}{2}+R_2 \sin^2 \frac{\alpha}{2}\right)}$$

2017 Bosnia and Herzegovina Regional 9.4
It is given isosceles triangle $ABC$ ($AB=AC$) such that $\angle BAC=108^{\circ}$. Angle bisector of angle $\angle ABC$ intersects side $AC$ in point $D$, and point $E$ is on side $BC$ such that $BE=AE$. If $AE=m$, find $ED$

2017 Bosnia and Herzegovina Regional 10.2
It is given triangle $ABC$. Let internal and external angle bisector of angle $\angle BAC$ intersect line $BC$ in points $D$ and $E$, respectively, and circumcircle of triangle $ADE$ intersects line $AC$ in point $F$. Prove that $FD$ is angle bisector of $\angle BFC$

2017 Bosnia and Herzegovina Regional 11.2
Let $ABC$ be an isosceles triangle such that $AB=AC$. Find angles of triangle $ABC$ if $\frac{AB}{BC}=1+2\cos{\frac{2\pi}{7}}$

2017 Bosnia and Herzegovina Regional 12.2
In triangle $ABC$ on side $AC$ are points $K$, $L$ and $M$ such that $BK$ is an angle bisector of $\angle ABL$, $BL$ is an angle bisector of $\angle KBM$ and $BM$ is an angle bisector of $\angle LBC$, respectively. Prove that $4 \cdot LM <AC$ and $3\cdot \angle BAC - \angle ACB < 180^{\circ}$

2018 Bosnia and Herzegovina Regional 9.5
Let $H$ be an orhocenter of an acute triangle $ABC$ and $M$ midpoint of side $BC$. If $D$ and $E$ are foots of perpendicular of $H$ on internal and external angle bisector of angle $\angle BAC$, prove that $M$, $D$ and $E$ are collinear

2018 Bosnia and Herzegovina Regional 10.4
Let $P$ be a point on circumcircle of triangle $ABC$ on arc  $\stackrel{\frown}{BC}$ which does not contain point $A$. Let lines $AB$ and $CP$ intersect at point $E$, and lines $AC$ and $BP$ intersect at $F$. If perpendicular bisector of side $AB$ intersects $AC$ in point $K$, and perpendicular bisector of side $AC$ intersects side $AB$ in point $J$, prove that:
${\left(\frac{CE}{BF}\right)}^2=\frac{AJ\cdot JE}{AK \cdot KF}$

2018 Bosnia and Herzegovina Regional 11.3
In triangle $ABC$ given is point $P$ such that $\angle ACP = \angle ABP = 10^{\circ}$, $\angle CAP = 20^{\circ}$ and $\angle BAP = 30^{\circ}$. Prove that $AC=BC$

2018 Bosnia and Herzegovina Regional 12.4
Let $ABCD$ be a cyclic quadrilateral and let $k_1$ and $k_2$ be circles inscribed in triangles $ABC$ and $ABD$. Prove that external common tangent of those circles (different from $AB$) is parallel with $CD$.