Bosnia & Herzegovina MO / TST 1996 - 2018 (IMO - EGMO) 46p

geometry problems from Math Olympiad of Bosnia and Herzegovina, named as Team Selection Tests (TST) inside aops,  with aops links in the names                                                             

collected inside aops here

IMO TST 1996 - 2017  

1996 Bosnia and Herzegovina TST P3

Let $M$ be a point inside quadrilateral $ABCD$ such that $ABMD$ is parallelogram. If $\angle CBM = \angle CDM$ prove that $\angle ACD = \angle BCM$

1997 Bosnia and Herzegovina TST P2

In isosceles triangle $ABC$ with base side $AB$, on side $BC$ it is given point $M$. Let $O$ be a circumcenter and $S$ incenter of triangle $ABC$. Prove that $$SM \mid \mid AC \Leftrightarrow OM \perp BS$$

1997 Bosnia and Herzegovina TST P4

b) Tetrahedron $ABCD$ has three pairs of equal opposing sides. Find length of height of tetrahedron in function od lengths of sides

1998 Bosnia and Herzegovina TST P1

Angle bisectors of angles by vertices $A$, $B$ and $C$ in triangle $ABC$ intersect opposing sides in points $A_1$, $B_1$ and $C_1$, respectively. Let $M$ be an arbitrary point on one of the lines $A_1B_1$, $B_1C_1$ and $C_1A_1$. Let $M_1$, $M_2$ and $M_3$ be orthogonal projections of point $M$ on lines $BC$, $CA$ and $AB$, respectively. Prove that one of the lines $MM_1$, $MM_2$ and $MM_3$ is equal to sum of other two.

1998 Bosnia and Herzegovina TST P4

Circle $k$  with radius $r$ touches the line $p$ in point $A$. Let $AB$ be a dimeter of circle and $C$ an arbitrary point of circle distinct from points $A$ and $B$. Let $D$ be a foot of perpendicular from point $C$ to line $AB$. Let $E$ be a point on extension of line $CD$, over point $D$, such that $ED=BC$. Let tangents on circle from point $E$ intersect line $p$ in points $K$ and $N$. Prove that length of $KN$ does not depend from $C$

1999 Bosnia and Herzegovina TST P4

Let angle bisectors of angles $\angle BAC$ and $\angle ABC$ of triangle $ABC$ intersect sides $BC$ and $AC$ in points $D$ and $E$, respectively. Let points $F$ and $G$ be foots of perpendiculars from point $C$ on lines $AD$ and $BE$, respectively. Prove that $FG \mid \mid AB$

2000 Bosnia and Herzegovina TST P2

Let $S$ be a point inside triangle $ABC$ and let lines $AS$, $BS$ and $CS$ intersect sides $BC$, $CA$ and $AB$ in points $X$, $Y$ and $Z$, respectively. Prove that $$\frac{BX\cdot CX}{AX^2}+\frac{CY\cdot AY}{BY^2}+\frac{AZ\cdot BZ}{CZ^2}=\frac{R}{r}-1$$ iff $S$ is incenter of $ABC$

2000 Bosnia and Herzegovina TST P6

It is given triangle $ABC$ such that $\angle ABC = 3 \angle CAB$. On side $AC$ there are two points $M$ and $N$ in order $A - N - M - C$ and $\angle CBM = \angle MBN = \angle NBA$. Let $L$ be an arbitrary point on side $BN$ and $K$ point on $BM$ such that $LK \mid \mid AC$. Prove that lines $AL$, $NK$ and $BC$ are concurrent.

2001 Bosnia and Herzegovina TST P1

On circle there are points $A$, $B$ and $C$ such that they divide circle in ratio $3:5:7$. Find angles of triangle $ABC$

2001 Bosnia and Herzegovina TST P4

In plane there  are two circles with radiuses $r_1$ and $r_2$, one outside the other. There are two external common tangents on those circles and one internal common tangent. The internal one intersects external ones in points $A$ and $B$ and touches one of the circles in point $C$. Prove that $AC \cdot BC=r_1\cdot r_2$

2002 Bosnia and Herzegovina TST P2

Triangle $ABC$ is given in a plane. Draw the bisectors of all three of its angles. Then draw the line that connects the points where the bisectors of angles $ABC$ and $ACB$ meet the opposite sides of the triangle. Through the point of intersection of this line and the bisector of angle $BAC$, draw another line parallel to $BC$. Let this line intersect $AB$ in $M$ and $AC$ in $N$. Prove that $2MN = BM+CN$.

2002 Bosnia and Herzegovina TST P5

The vertices of the convex quadrilateral $ABCD$ and the intersection point $S$ of its diagonals are integer points in the plane. Let $P$ be the area of $ABCD$ and $P_1$ the area of triangle $ABS$. Prove that $\sqrt{P} \ge \sqrt{P_1}+\frac{\sqrt2}2$

2003 Bosnia and Herzegovina TST P2
Upon sides $AB$ and $BC$ of triangle $ABC$ are constructed squares $ABB_{1}A_{1}$ and $BCC_{1}B_{2}$. Prove that lines $AC_{1}$, $CA_{1}$ and altitude from $B$ to side $AC$ are concurrent.

2003 Bosnia and Herzegovina TST P4
In triangle $ABC$ $AD$ and $BE$ are altitudes. Let $L$ be a point on $ED$ such that $ED$ is orthogonal to $BL$. If $LB^2=LD\cdot LE$ prove that triangle $ABC$ is isosceles.

2004 Bosnia and Herzegovina TST P1
Circle $k$ with center $O$ is touched from inside by two circles in points $S$ and $T,$ respectively. Let those two circles intersect at points $M$ and $N$, such that $N$ is closer to line $ST$. Prove that $OM$ and $MN$ are perpendicular iff $S$, $N$ and $T$ are collinear.

2004 Bosnia and Herzegovina TST P6 

It is given triangle $ABC$ and parallelogram $ASCR$ with diagonal $AC$. Let line constructed through point $B$ parallel with $CS$ intersects line $AS$ and $CR$ in $M$ and $P$, respectively. Let line constructed through point $B$ parallel with $AS$ intersects line $AR$ and $CS$ in $N$ and $Q$, respectively. Prove that lines $RS$, $MN$ and $PQ$ are concurrent.

2005 Bosnia and Herzegovina TST P1

Let $H$ be an orthocenter of an acute triangle $ABC$. Prove that midpoints of $AB$ and $CH$ and intersection point of angle bisectors of $\angle CAH$ and $\angle CBH$ lie on the same line.

2005 Bosnia and Herzegovina TST P4

On the line which contains diameter $PQ$ of circle $k(S,r)$, point $A$ is chosen outside the circle such that tangent $t$ from point $A$ touches the circle in point $T$. Tangents on circle $k$ in points $P$ and $Q$ are $p$ and $q$, respectively. If $PT \cap q={N}$ and $QT \cap p={M}$, prove that points $A$, $M$ and $N$ are collinear.

2006 Bosnia and Herzegovina TST P2

It is given a triangle $\triangle ABC$. Determine the locus of center of rectangle inscribed in triangle $ABC$ such that one side of rectangle lies on side $AB$.

2006 Bosnia and Herzegovina TST P5

Triangle $ABC$ is inscribed in circle with center $O$. Let $P$ be a point on arc $AB$ which does not contain point $C$. Perpendicular from point $P$ on line $BO$ intersects side $AB$ in point $S$, and side $BC$ in $T$. Perpendicular from point $P$ on line $AO$ intersects side $AB$ in point $Q$, and side $AC$ in $R$.
i) Prove that triangle $PQS$ is isosceles
ii) Prove that $\frac{PQ}{QR}=\frac{ST}{PQ}$

2007 Bosnia and Herzegovina TST P1

Let $ABC$ be a triangle such that length of internal angle bisector from $B$ is equal to $s$. Also, length of external angle bisector from $B$ is equal to $s_1$. Find area of triangle $ABC$ if $\frac{AB}{BC} = k$

2007 Bosnia and Herzegovina TST P5

Triangle $ABC$ is right angled such that $\angle ACB=90^{\circ}$ and $\frac {AC}{BC} = 2$. Let the line parallel to side $AC$ intersects line segments $AB$ and $BC$ in $M$ and $N$ such that $\frac {CN}{BN} = 2$. Let $O$ be the intersection point of lines $CM$ and $AN$. On segment $ON$ lies point $K$ such that $OM+OK=KN$. Let $T$ be the intersection point of angle bisector of $\angle ABC$ and line from $K$ perpendicular to $AN$. Determine value of  $\angle MTB$.

2008 Bosnia and Herzegovina TST P1

Prove that in an isosceles triangle  $\triangle ABC$ with $AC=BC=b$ following inequality holds $b> \pi r$, where $r$ is inradius.

2008 Bosnia and Herzegovina TST P5

Let $AD$ be height of triangle $\triangle ABC$ and $R$ circumradius. Denote by $E$ and $F$ feet of perpendiculars from point $D$ to sides $AB$ and $AC$. If $AD=R\sqrt{2}$, prove that circumcenter of triangle $\triangle ABC$ lies on line $EF$.

2009 Bosnia and Herzegovina TST P1

Denote by $M$ and $N$ feets of perpendiculars from $A$ to angle bisectors of exterior angles at $B$ and $C,$ in triangle $\triangle ABC.$ Prove that the length of segment $MN$ is equal to semiperimeter of triangle $\triangle ABC.$

2009 Bosnia and Herzegovina TST P5

Line $p$ intersects sides $AB$ and $BC$ of triangle $\triangle ABC$ at points $M$ and $K.$ If area of triangle $\triangle MBK$ is equal to area of quadrilateral $AMKC,$ prove that $\frac{\left|MB\right|+\left|BK\right|}{\left|AM\right|+\left|CA\right|+\left|KC\right|}\geq\frac{1}{3}$

2010 Bosnia and Herzegovina TST P2

Let $AB$ and $FD$ be chords in circle, which does not intersect and $P$ point on arc $AB$ which does not contain chord $FD$. Lines $PF$ and $PD$ intersect chord $AB$ in $Q$ and $R$. Prove that $\frac{AQ \cdot RB}{QR}$ is constant, while point $P$ moves along the ray $AB$.

2010 Bosnia and Herzegovina TST P4

Convex quadrilateral is divided by diagonals into four triangles with congruent inscribed circles. Prove that this quadrilateral is rhombus.

2011 Bosnia and Herzegovina TST P1

In triangle $ABC$ it holds $|BC|= \frac{1}{2}(|AB|+|AC|)$. Let $M$ and $N$ be midpoints of $AB$ and $AC$, and let $I$ be the incenter of $ABC$. Prove that $A, M, I, N$ are concyclic.

2011 Bosnia and Herzegovina TST P6

In quadrilateral $ABCD$ sides $AD$ and $BC$ aren't parallel. Diagonals $AC$ and $BD$ intersect in $E$. $F$ and $G$ are points on sides $AB$ and $DC$ such $\frac{AF}{FB}=\frac{DG}{GC}=\frac{AD}{BC}$ Prove that if $E, F, G$ are collinear then $ABCD$ is cyclic.

2012 Bosnia and Herzegovina TST P1

Let $D$ be the midpoint of the arc $B-A-C$ of the circumcircle of $\triangle ABC (AB<AC)$. Let $E$ be the foot of perpendicular from $D$ to $AC$. Prove that $|CE|=\frac{|BA|+|AC|}{2}$.

2012 Bosnia and Herzegovina TST P5

Given is a triangle $\triangle ABC$ and points $M$ and $K$ on lines $AB$ and $CB$ such that $AM=AC=CK$. Prove that the length of the radius of the circumcircle of triangle $\triangle BKM$ is equal to the lenght $OI$, where $O$ and $I$ are centers of the circumcircle and the incircle of $\triangle ABC$, respectively. Also prove that $OI\perp MK$.

2013 Bosnia and Herzegovina TST P1

Triangle $ABC$ is right angled at $C$. Lines $AM$ and $BN$ are internal angle bisectors. $AM$ and $BN$ intersect altitude $CH$ at points $P$ and $Q$ respectively. Prove that the line which passes through the midpoints of segments $QN$ and $PM$ is parallel to $AB$.

2013 Bosnia and Herzegovina TST P6

In triangle $ABC$, $I$ is the incenter. We have chosen points $P,Q,R$ on segments $IA,IB,IC$ respectively such that $IP\cdot IA=IQ \cdot IB=IR\cdot IC$. Prove that the points $I$ and $O$ belong to Euler line of triangle $PQR$ where $O$ is circumcenter of $ABC$.

2014 Bosnia and Herzegovina TST P1

Let $k$ be the circle and $A$ and $B$ points on circle which are not diametrically opposite. On minor arc $AB$ lies point arbitrary point $C$. Let $D$, $E$ and $F$ be foots of perpendiculars from $C$ on chord $AB$ and tangents of circle $k$ in points $A$ and $B$. Prove that $CD= \sqrt {CE \cdot CF}$

2014 Bosnia and Herzegovina TST P6

Let $D$ and $E$ be foots of altitudes from $A$ and $B$ of triangle $ABC$, $F$ be intersection point of angle bisector from $C$ with side $AB$, and $O$, $I$ and $H$ be circumcenter, center of inscribed circle and orthocenter of triangle $ABC$, respectively. If $\frac{CF}{AD}+ \frac{CF}{BE}=2$, prove that $OI = IH$.

2015 Bosnia and Herzegovina TST P2

Let $D$ be an arbitrary point on side $AB$ of triangle $ABC$. Circumcircles of triangles $BCD$ and $ACD$ intersect sides $AC$ and $BC$ at points $E$ and $F$, respectively. Perpendicular bisector of $EF$ cuts $AB$ at point $M$, and line perpendicular to $AB$ at $D$ at point $N$. Lines $AB$ and $EF$ intersect at point $T$, and the second point of intersection of circumcircle of triangle $CMD$ and line $TC$ is $U$. Prove that $NC=NU$

2015 Bosnia and Herzegovina TST P6

Let $D$, $E$ and $F$ be points in which incircle of triangle $ABC$ touches sides $BC$, $CA$ and $AB$, respectively, and let $I$ be a center of that circle.Furthermore, let  $P$ be a foot of perpendicular from point $I$ to line $AD$, and let $M$ be midpoint of $DE$. If $\{N\}=PM\cap{AC}$, prove that $PN \parallel EF$

2016 Bosnia and Herzegovina TST P1

Let $ABCD$ be a quadrilateral inscribed in circle $k$. Lines $AB$ and $CD$ intersect at point $E$ such that $AB=BE$. Let $F$ be the intersection point of tangents on circle $k$ in points $B$ and $D$, respectively. If the lines $AB$ and $DF$ are parallel, prove that $A$, $C$ and $F$ are collinear.

2016 Bosnia and Herzegovina TST P5

Let $k$ be a circumcircle of triangle $ABC$ $(AC<BC)$. Also, let $CL$ be an angle bisector of angle $ACB$ $(L \in AB)$, $M$ be a midpoint of arc $AB$ of circle $k$ containing the point $C$, and let $I$ be an incenter of a triangle $ABC$. Circle $k$ cuts line $MI$ at point $K$ and circle with diameter $CI$ at $H$. If the circumcircle of triangle $CLK$ intersects $AB$ again at $T$, prove that $T$, $H$ and $C$ are collinear.

2017 Bosnia and Herzegovina TST P1

Incircle of triangle $ABC$ touches $AB,AC$ at $P,Q$. $BI, CI$ intersect with $PQ$ at $K,L$. Prove that circumcircle of $ILK$ is tangent to incircle of $ABC$ if and only if $AB+AC=3BC$

2017 Bosnia and Herzegovina TST P6
Given is an acute triangle $ABC$. $M$ is an arbitrary point at the side $AB$ and $N$ is midpoint of $AC$. The foots of the perpendiculars from $A$ to $MC$ and $MN$ are points $P$ and $Q$. Prove that center of the circumcircle of triangle $PQN$ lies on the fixed line for all points $M$ from the side $AB$.

2018 Bosnia and Herzegovina TST P1
In acute triangle $ABC$ $(AB < AC)$ let $D$, $E$ and $F$ be foots of perpedicular from $A$, $B$ and $C$ to $BC$, $CA$ and $AB$, respectively. Let $P$ and $Q$ be points on line $EF$ such that $DP \perp EF$ and $BQ=CQ$. Prove that $\angle ADP = \angle PBQ$

EGMO TST 2017-19

2017 Bosnia and Herzegovina  EGMO TST P2
It is given triangle $ABC$ and points $P$ and $Q$ on sides $AB$ and $AC$, respectively, such that $PQ\mid\mid BC$. Let $X$ and $Y$ be intersection points of lines $BQ$ and $CP$ with circumcircle $k$ of triangle $APQ$, and $D$ and $E$ intersection points of lines $AX$ and $AY$ with side $BC$. If $2\cdot DE=BC$, prove that circle $k$ contains intersection point of angle bisector of $\angle BAC$ with $BC$ 

2018 Bosnia and Herzegovina  EGMO TST P3
Let $O$ be a circumcenter of acute triangle $ABC$ and let $O_1$ and $O_2$ be circumcenters of triangles $OAB$ and $OAC$, respectively. Circumcircles of triangles $OAB$ and $OAC$ intersect side $BC$ in points $D$ ($D \neq B$) and $E$ ($E \neq C$), respectively. Perpendicular bisector of side $BC$ intersects side $AC$ in point $F$($F \neq A$). Prove that circumcenter of triangle $ADE$ lies on $AC$ iff $F$ lies on line $O_1O_2$

2019 Bosnia and Herzegovina  EGMO TST P3
The circle inscribed in the triangle $ABC$  touches the sides $AB$  and  $AC$ at the points $K$  and $L$ , respectively. The angle bisectors from $B$ and $C$ intersect the altitude of the triangle from the vertice $A$ at the points $Q$  and $R$ , respectively. Prove that one of the points of intersection of the circles circumscribed around the triangles  $BKQ$   and   $CPL$ lies on $BC$.