geometry problems from Junior Balkan Mathematical Olympiads Team Selection Tests of Bosnia and Herzegovina (JBMO TST) with aops links
2003 Bosnia and Herzegovina JBMO TST P4
In the trapezoid $ABCD$ ($AB \parallel DC$) the bases have lengths $a$ and $c$ ($c < a$), while the other sides have lengths $b$ and $d$. The diagonals are of lengths $m$ and $n$. It is known that $m^2 + n^2 = (a + c)^2$.
a) Find the angle between the diagonals of the trapezoid.
b) Prove that $a + c < b + d$.
c) Prove that $ac < bd$.
Point $ M$ is given in the interior of parallelogram $ ABCD$, and the point $ N$ inside triangle $ AMD$ is chosen so that $ < MNA + < MCB = MND + < MBC = 180^0$. Prove that $ MN$ is parallel to $ AB$.
2009 Bosnia and Herzegovina JBMO TST P1
Lengths of sides of triangle $ABC$ are positive integers, and smallest side is equal to $2$. Determine the area of triangle $P$ if $v_c = v_a + v_b$, where $v_a$, $v_b$ and $v_c$ are lengths of altitudes in triangle $ABC$ from vertices $A$, $B$ and $C$, respectively.
2010 Bosnia and Herzegovina JBMO TST P3
Points $M$ and $N$ are given on sides $AD$ and $BC$ of rhombus $ABCD$, respectively. Line $MC$ intersects line $BD$ in point $T$, line $MN$ intersects line $BD$ in point $U$, line $CU$ intersects line $AB$ in point $Q$ and line $QT$ intersects line $CD$ in $P$. Prove that triangles $QCP$ and $MCN$ have equal area.
2011 Bosnia and Herzegovina JBMO TST P3
In isosceles triangle $ABC$ ($AC=BC$), angle bisector $\angle BAC$ and altitude $CD$ from point $C$ intersect at point $O$, such that $CO=3 \cdot OD$. In which ratio does altitude from point $A$ on side $BC$ divide altitude $CD$ of triangle $ABC$
2012 Bosnia and Herzegovina JBMO TST P1
On circle $k$ there are clockwise points $A$, $B$, $C$, $D$ and $E$ such that $\angle ABE = \angle BEC = \angle ECD = 45^{\circ}$. Prove that $AB^2 + CE^2 = BE^2 + CD^2$
2013 Bosnia and Herzegovina JBMO TST P3
Let $M$ and $N$ be touching points of incircle with sides $AB$ and $AC$ of triangle $ABC$, and $P$ intersection point of line $MN$ and angle bisector of $\angle ABC$. Prove that $\angle BPC =90 ^{\circ}$
2014 Bosnia and Herzegovina JBMO TST P2
In triangle $ABC$, on line $CA$ it is given point $D$ such that $CD = 3 \cdot CA$ (point $A$ is between points $C$ and $D$), and on line $BC$ it is given point $E$ ($E \neq B$) such that $CE=BC$. If $BD=AE$, prove that $\angle BAC= 90^{\circ}$
2015 Bosnia and Herzegovina JBMO TST P3
Let $AD$ be an altitude of triangle $ABC$, and let $M$, $N$ and $P$ be midpoints of $AB$, $AD$ and $BC$, respectively. Furthermore let $K$ be a foot of perpendicular from point $D$ to line $AC$, and let $T$ be point on extension of line $KD$ (over point $D$) such that $\mid DT \mid = \mid MN \mid + \mid DK \mid$. If $\mid MP \mid = 2 \cdot \mid KN \mid$, prove that $\mid AT \mid = \mid MC \mid$.
[not in JBMO Shortlist]
collected inside aops here
2003-22
2003 Bosnia and Herzegovina JBMO TST P4
In the trapezoid $ABCD$ ($AB \parallel DC$) the bases have lengths $a$ and $c$ ($c < a$), while the other sides have lengths $b$ and $d$. The diagonals are of lengths $m$ and $n$. It is known that $m^2 + n^2 = (a + c)^2$.
a) Find the angle between the diagonals of the trapezoid.
b) Prove that $a + c < b + d$.
c) Prove that $ac < bd$.
2004 Bosnia and Herzegovina JBMO TST P4
Let $ABCD$ be a parallelogram. On the ray $(DB$ a point $E$ is given such that the ray $(AB$ is the angle bisector of $\angle CAE$. Let $F$ be the intersection of $CE$ and $AB$. Prove that $\frac{AB}{BF} - \frac{AC}{AE} = 1$
Let $ABCD$ be a parallelogram. On the ray $(DB$ a point $E$ is given such that the ray $(AB$ is the angle bisector of $\angle CAE$. Let $F$ be the intersection of $CE$ and $AB$. Prove that $\frac{AB}{BF} - \frac{AC}{AE} = 1$
2004 Bosnia and Herzegovina JBMO TST P5
In the isosceles triangle $ABC$ ($AC = BC$), $AB =\sqrt3$ and the altitude $CD =\sqrt2$. Let E and F be the midpoints of $BC$ and $DB$, respectively and $G$ be the intersection of $AE$ and $CF$. Prove that $D$ belongs to the angle bisector of $\angle AGF$.
In the isosceles triangle $ABC$ ($AC = BC$), $AB =\sqrt3$ and the altitude $CD =\sqrt2$. Let E and F be the midpoints of $BC$ and $DB$, respectively and $G$ be the intersection of $AE$ and $CF$. Prove that $D$ belongs to the angle bisector of $\angle AGF$.
2005 Bosnia and Herzegovina JBMO TST P4
The sum of the angles on the bigger base of a trapezoid is $90^o$. Prove that the line segment whose ends are the midpoints of the bases, is equal to the line segment whose ends are the midpoints of the diagonals.
The sum of the angles on the bigger base of a trapezoid is $90^o$. Prove that the line segment whose ends are the midpoints of the bases, is equal to the line segment whose ends are the midpoints of the diagonals.
2006 Bosnia and Herzegovina JBMO TST P2
In an acute triangle $ABC, C = 60^o$. If $AA'$ and $BB'$ are two of the altitudes and $C_1$ is the midpoint of $AB$, prove that triangle $C_1A'B'$ is equilateral.
In an acute triangle $ABC, C = 60^o$. If $AA'$ and $BB'$ are two of the altitudes and $C_1$ is the midpoint of $AB$, prove that triangle $C_1A'B'$ is equilateral.
2007 Bosnia and Herzegovina JBMO TST P4
Let $I$ be the incenter of the triangle $ABC$ ($AB < BC$). Let $M$ be the midpoint of $AC$, and let $N$ be the midpoint of the arc $AC$ of the circumcircle of $ABC$ which contains $B$. Prove that $\angle IMA = \angle INB$.
2008 Bosnia and Herzegovina JBMO TST P3Let $I$ be the incenter of the triangle $ABC$ ($AB < BC$). Let $M$ be the midpoint of $AC$, and let $N$ be the midpoint of the arc $AC$ of the circumcircle of $ABC$ which contains $B$. Prove that $\angle IMA = \angle INB$.
Point $ M$ is given in the interior of parallelogram $ ABCD$, and the point $ N$ inside triangle $ AMD$ is chosen so that $ < MNA + < MCB = MND + < MBC = 180^0$. Prove that $ MN$ is parallel to $ AB$.
Lengths of sides of triangle $ABC$ are positive integers, and smallest side is equal to $2$. Determine the area of triangle $P$ if $v_c = v_a + v_b$, where $v_a$, $v_b$ and $v_c$ are lengths of altitudes in triangle $ABC$ from vertices $A$, $B$ and $C$, respectively.
Points $M$ and $N$ are given on sides $AD$ and $BC$ of rhombus $ABCD$, respectively. Line $MC$ intersects line $BD$ in point $T$, line $MN$ intersects line $BD$ in point $U$, line $CU$ intersects line $AB$ in point $Q$ and line $QT$ intersects line $CD$ in $P$. Prove that triangles $QCP$ and $MCN$ have equal area.
2011 Bosnia and Herzegovina JBMO TST P3
In isosceles triangle $ABC$ ($AC=BC$), angle bisector $\angle BAC$ and altitude $CD$ from point $C$ intersect at point $O$, such that $CO=3 \cdot OD$. In which ratio does altitude from point $A$ on side $BC$ divide altitude $CD$ of triangle $ABC$
2012 Bosnia and Herzegovina JBMO TST P1
On circle $k$ there are clockwise points $A$, $B$, $C$, $D$ and $E$ such that $\angle ABE = \angle BEC = \angle ECD = 45^{\circ}$. Prove that $AB^2 + CE^2 = BE^2 + CD^2$
Let $M$ and $N$ be touching points of incircle with sides $AB$ and $AC$ of triangle $ABC$, and $P$ intersection point of line $MN$ and angle bisector of $\angle ABC$. Prove that $\angle BPC =90 ^{\circ}$
2014 Bosnia and Herzegovina JBMO TST P2
In triangle $ABC$, on line $CA$ it is given point $D$ such that $CD = 3 \cdot CA$ (point $A$ is between points $C$ and $D$), and on line $BC$ it is given point $E$ ($E \neq B$) such that $CE=BC$. If $BD=AE$, prove that $\angle BAC= 90^{\circ}$
2015 Bosnia and Herzegovina JBMO TST P3
Let $AD$ be an altitude of triangle $ABC$, and let $M$, $N$ and $P$ be midpoints of $AB$, $AD$ and $BC$, respectively. Furthermore let $K$ be a foot of perpendicular from point $D$ to line $AC$, and let $T$ be point on extension of line $KD$ (over point $D$) such that $\mid DT \mid = \mid MN \mid + \mid DK \mid$. If $\mid MP \mid = 2 \cdot \mid KN \mid$, prove that $\mid AT \mid = \mid MC \mid$.
2016 Bosnia and Herzegovina JBMO TST P3
Let $O$ be a center of circle which passes through vertices of quadrilateral $ABCD$, which has perpendicular diagonals. Prove that sum of distances of point $O$ to sides of quadrilateral $ABCD$ is equal to half of perimeter of $ABCD$.
Let $O$ be a center of circle which passes through vertices of quadrilateral $ABCD$, which has perpendicular diagonals. Prove that sum of distances of point $O$ to sides of quadrilateral $ABCD$ is equal to half of perimeter of $ABCD$.
2017 Bosnia and Herzegovina JBMO TST P3
Let $ABC$ be a triangle such that $\angle ABC = 90 ^{\circ}$. Let $I$ be an incenter of $ABC$ and let $F$, $D$ and $E$ be points where incircle touches sides $AB$, $BC$ and $AC$, respectively. If lines $CI$ and $EF$ intersect at point $M$ and if $DM$ and $AB$ intersect in $N$, prove that $AI=ND$
Let $ABC$ be a triangle such that $\angle ABC = 90 ^{\circ}$. Let $I$ be an incenter of $ABC$ and let $F$, $D$ and $E$ be points where incircle touches sides $AB$, $BC$ and $AC$, respectively. If lines $CI$ and $EF$ intersect at point $M$ and if $DM$ and $AB$ intersect in $N$, prove that $AI=ND$
2018 Bosnia and Herzegovina JBMO TST P3
Let $\Gamma$ be circumscribed circle of triangle $ABC$ ($AB \ne AC$). Let $O$ be circumcenter of triangle $ABC$. Let $M$ be a point where angle bisector of angle $BAC$ intersects $\Gamma$ . Let $D$ ($D\ne M$) be a point where circumscribed circle of triangle $BOM$ intersects line segment $AM$ and let $E$ ($E\ne M$) be a point where circumscribed circle of triangle $COM$ intersects line segment $AM$. Prove that $BD+CE=AM$
2019 Bosnia and Herzegovina JBMO TST P2
Let $ABC$ be a triangle and $AD$ the angle bisector ($D\in BC$). The perpendicular from $B$ to $AD$ cuts the circumcircle of triangle $ABD$ at $E$. If $O$ is the center of the circle around $ABC$ , prove $A,O,E$ are collinear.
2020 Bosnia and Herzegovina JBMO TST P3
Let $\Gamma$ be circumscribed circle of triangle $ABC$ ($AB \ne AC$). Let $O$ be circumcenter of triangle $ABC$. Let $M$ be a point where angle bisector of angle $BAC$ intersects $\Gamma$ . Let $D$ ($D\ne M$) be a point where circumscribed circle of triangle $BOM$ intersects line segment $AM$ and let $E$ ($E\ne M$) be a point where circumscribed circle of triangle $COM$ intersects line segment $AM$. Prove that $BD+CE=AM$
2019 Bosnia and Herzegovina JBMO TST P2
Let $ABC$ be a triangle and $AD$ the angle bisector ($D\in BC$). The perpendicular from $B$ to $AD$ cuts the circumcircle of triangle $ABD$ at $E$. If $O$ is the center of the circle around $ABC$ , prove $A,O,E$ are collinear.
2020 Bosnia and Herzegovina JBMO TST P3
The angle bisector of $\angle ABC$ of triangle $ABC$ ($AB>BC$) cuts the circumcircle of that triangle in $K$. The foot of the perpendicular from $K$ to $AB$ is $N$, and $P$ is the midpoint of $BN$. The line through $P$ parallel to $BC$ cuts line $BK$ in $T$. Prove that the line $NT$ passes through the midpoint of $AC$.
sources: dms.rs , https://pregatirematematicaolimpiadejuniori.wordpress.com/
In the convex quadrilateral $ABCD$, $AD = BD$ and $\angle ACD = 3 \angle BAC$. Let $M$ be the midpoint of side $AD$. If the lines $CM$ and $AB$ are parallel, prove that the angle $\angle ACB$ is right.
Let $ABC$ be an acute triangle. Tangents on the circumscribed circle of triangle $ABC$ at points $B$ and $C$ intersect at point $T$. Let $D$ and $E$ be a foot of the altitudes from $T$ onto $AB$ and $AC$ and let $M$ be the midpoint of $BC$. Prove:
a) Prove that $M$ is the orthocenter of the triangle $ADE$.
b) Prove that $TM$ cuts $DE$ in half.
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