geometry problems from Saudi Arabian Junior Balkan Mathematical Olympiads Team Selection Tests (JBMO TST) with aops links
A right triangle $ABC$ with $\angle C=90^o$ is inscribed in a circle. Suppose that $K$ is the midpoint of the arc $BC$ that does not contain $A$. Let $N$ be the midpoint of the segment $AC$, and $M$ be the intersection point of the ray $KN$ and the circle.The tangents to the circle drawn at $A$ and $C$ meet at $E$. prove that $\angle EMK = 90^o$
2015 Saudi Arabia JBMO TST 2.4
Let $ABC$ be a right triangle with the hypotenus $BC.$ Let $BE$ be the bisector of the angle $\angle ABC.$ The circumcircle of the triangle $BCE$ cuts the segment $AB$ again at $F.$ Let $K$ be the projection of $A$ on $BC.$ The point $L$ lies on the segment $AB$ such that $BL=BK.$ Prove that $\frac{AL}{AF}=\sqrt{\frac{BK}{BC}}.$
2015 Saudi Arabia JBMO TST 3.3
Let $ABC$ be an acute-angled triangle inscribed in the circle $(O)$. Let $AD$ be the diameter of $(O)$. The points $M,N$ are chosen on $BC$ such that $OM\parallel AB, ON\parallel AC$. The lines $DM,DN$ cut $(O)$ again at $P,Q$. Prove that $BC=DP=DQ$.
2017 Saudi Arabia JBMO TST 1.4
Let $ABC$ be an acute, non isosceles triangle and $(O)$ be its circumcircle (with center $O$). Denote by $G$ the centroid of the triangle $ABC$, by $H$ the foot of the altitude from $A$ onto the side $BC$ and by $I$ the midpoint of $AH$. The line $IG$ intersects $BC$ at $K$.
1. Prove that $CK = BH$.
2. The ray $(GH$ intersects $(O)$ at L. Denote by $T$ the circumcenter of the triangle $BHL$. Prove that $AO$ and $BT$ intersect on the circle $(O)$.
2017 Saudi Arabia JBMO TST 2.3
Let $(O)$ be a circle, and $BC$ be a chord of $(O)$ such that $BC$ is not a diameter. Let $A$ be a point on the larger arc $BC$ of $(O)$, and let $E, F$ be the feet of the perpendiculars from $B$ and $C$ to $AC$ and $AB$, respectively.
1. Prove that the tangents to $(AEF)$ at $E$ and $F$ intersect at a fixed point $M$ when $A$ moves on the larger arc $BC$ of $(O)$.
2. Let $T$ be the intersection of $EF$ and $BC$, and let $H$ be the orthocenter of $ABC$. Prove that $TH$ is perpendicular to $AM$.
2017 Saudi Arabia JBMO TST 3.3
Let $BC$ be a chord of a circle $(O)$ such that $BC$ is not a diameter. Let $AE$ be the diameter perpendicular to $BC$ such that $A$ belongs to the larger arc $BC$ of $(O)$. Let $D$ be a point on the larger arc $BC$ of $(O)$ which is different from $A$. Suppose that $AD$ intersects $BC$ at $S$ and $DE$ intersects $BC$ at $T$. Let $F$ be the midpoint of $ST$ and $I$ be the second intersection point of the circle $(ODF)$ with the line $BC$.
1. Let the line passing through $I$ and parallel to $OD$ intersect $AD$ and $DE$ at $M$ and $N$, respectively. Find the maximum value of the area of the triangle $MDN$ when $D$ moves on the larger arc $BC$ of $(O)$ (such that $D \ne A$).
2. Prove that the perpendicular from $D$ to $ST$ passes through the midpoint of $MN$
Let $ABC$ be a triangle inscribed in the circle $(O)$, with orthocenter $H$. Let d be an arbitrary line which passes through $H$ and intersects $(O)$ at $P$ and $Q$. Draw diameter $AA'$ of circle $(O)$. Lines $A'P$ and $A'Q$ meet $BC$ at $K$ and $L$, respectively. Prove that $O, K, L$ and $A'$ are concyclic.
[not in JBMO Shortlist]
collected inside aops here
2015, 2017-19, 2021-22
2015 Saudi Arabia JBMO TST 1.3A right triangle $ABC$ with $\angle C=90^o$ is inscribed in a circle. Suppose that $K$ is the midpoint of the arc $BC$ that does not contain $A$. Let $N$ be the midpoint of the segment $AC$, and $M$ be the intersection point of the ray $KN$ and the circle.The tangents to the circle drawn at $A$ and $C$ meet at $E$. prove that $\angle EMK = 90^o$
2015 Saudi Arabia JBMO TST 2.4
Let $ABC$ be a right triangle with the hypotenus $BC.$ Let $BE$ be the bisector of the angle $\angle ABC.$ The circumcircle of the triangle $BCE$ cuts the segment $AB$ again at $F.$ Let $K$ be the projection of $A$ on $BC.$ The point $L$ lies on the segment $AB$ such that $BL=BK.$ Prove that $\frac{AL}{AF}=\sqrt{\frac{BK}{BC}}.$
2015 Saudi Arabia JBMO TST 3.3
Let $ABC$ be an acute-angled triangle inscribed in the circle $(O)$. Let $AD$ be the diameter of $(O)$. The points $M,N$ are chosen on $BC$ such that $OM\parallel AB, ON\parallel AC$. The lines $DM,DN$ cut $(O)$ again at $P,Q$. Prove that $BC=DP=DQ$.
Let $ABC$ be an acute, non isosceles triangle and $(O)$ be its circumcircle (with center $O$). Denote by $G$ the centroid of the triangle $ABC$, by $H$ the foot of the altitude from $A$ onto the side $BC$ and by $I$ the midpoint of $AH$. The line $IG$ intersects $BC$ at $K$.
1. Prove that $CK = BH$.
2. The ray $(GH$ intersects $(O)$ at L. Denote by $T$ the circumcenter of the triangle $BHL$. Prove that $AO$ and $BT$ intersect on the circle $(O)$.
2017 Saudi Arabia JBMO TST 2.3
Let $(O)$ be a circle, and $BC$ be a chord of $(O)$ such that $BC$ is not a diameter. Let $A$ be a point on the larger arc $BC$ of $(O)$, and let $E, F$ be the feet of the perpendiculars from $B$ and $C$ to $AC$ and $AB$, respectively.
1. Prove that the tangents to $(AEF)$ at $E$ and $F$ intersect at a fixed point $M$ when $A$ moves on the larger arc $BC$ of $(O)$.
2. Let $T$ be the intersection of $EF$ and $BC$, and let $H$ be the orthocenter of $ABC$. Prove that $TH$ is perpendicular to $AM$.
2017 Saudi Arabia JBMO TST 3.3
Let $BC$ be a chord of a circle $(O)$ such that $BC$ is not a diameter. Let $AE$ be the diameter perpendicular to $BC$ such that $A$ belongs to the larger arc $BC$ of $(O)$. Let $D$ be a point on the larger arc $BC$ of $(O)$ which is different from $A$. Suppose that $AD$ intersects $BC$ at $S$ and $DE$ intersects $BC$ at $T$. Let $F$ be the midpoint of $ST$ and $I$ be the second intersection point of the circle $(ODF)$ with the line $BC$.
1. Let the line passing through $I$ and parallel to $OD$ intersect $AD$ and $DE$ at $M$ and $N$, respectively. Find the maximum value of the area of the triangle $MDN$ when $D$ moves on the larger arc $BC$ of $(O)$ (such that $D \ne A$).
2. Prove that the perpendicular from $D$ to $ST$ passes through the midpoint of $MN$
Let $ABC$ be a triangle inscribed in circle $(O)$ such that points $B, C$ are fixed, while $A$ moves on major arc $BC$ of $(O)$. The tangents through $B$ and $C$ to $(O)$ intersect at $P$. The circle with diameter $OP$ intersects $AC$ and $AB$ at $D$ and $E$, respectively. Prove that $DE$ is tangent to a fixed circle whose radius is half the radius of $(O)$.
2018 Saudi Arabia JBMO TST 1.3
Let $ABC$ be a triangle inscribed in the circle $K_1$ and $I$ be center of the inscribed in $ABC$ circle. The lines $IB$ and $IC$ intersect circle $K_1$ again in $J$ and $L$. Circle $K_2$, circumscribed to $IBC$, intersects again $CA$ and $AB$ in $E$ and $F$. Show that $EL$ and $FJ$ intersects on the circle $K_2$.
2018 Saudi Arabia JBMO TST 2.4
Let $ABC$ be a acute triangle in which $O$ and $H$ are the center of the circumscribed circle, respectively the orthocenter. Let $M$ be a point on the small arc $BC$ of the circumscribed circle (different from $B$ and $C$) and be $D, E, F$ be the symmetrical of the point $M$ to the lines $OA, OB, OC$. We note with $K$ the intersection of $BF$ and $CE$ and $I$ is the center of the circle inscribed in the triangle $DEF$.
a) Show that the segment bisectors of the segments $EF$ and $IK$ intersect on the circle
circumscribed to triangle $ABC$.
a) Prove that points $H, K, I$ are collinear.
2018 Saudi Arabia JBMO TST 3.2
Let $ABCD$ be a square inscribed in circle $K$. Let $P$ be a point on the small arc $CD$ of circle $K$. The line $PB$ intersects $AC$ in $E$. The line $PA$ intersects $DB$ in $F$. The circle circumscribed to triangle $PEF$ intersects for second time $K$ in $Q$. Prove that $PQ$ is parallel to $CD$.
2019 Saudi Arabia JBMO TST 1.4
Let $AD$ be the perpendicular to the hypotenuse $BC$ of the right triangle $ABC$. Let $DE$ be the height of the triangle $ADB$ and $DZ$ be the height of the triangle $ADC$. On the line $AB$ is chosen the point $N$ so that $CN$ is parallel to $EZ$. Let $A'$ be symmetrical of $A$ to $EZ$ and $I, K$ projections of $A'$ on $AB$, respectively, on $AC$. Prove that $<$ $NA'T$ $=$ $<$ $ADT$, where $T$ is the point of intersection of $IK$ and $DE$.
2019 Saudi Arabia JBMO TST 2.1
On the sides $BC$ and $CD$ of the square $ABCD$ of side $1$, are chosen the points $E$, respectively $F$, so that $<$ $EAB$ $=$ $20$ . If $<$ $EAF$ $=$ $45$, calculate the distance from point $A$ to the line $EF$.
2019 Saudi Arabia JBMO TST 3.3
Let $ABC$ be an acute and scalene triangle. Points $D$ and $E$ are in the interior of the triangle so that $\angle DAB = \angle DCB$, $\angle DAC = \angle DBC$, $\angle EAB = \angle EBC$ and $\angle EAC = \angle ECB$. Prove that the triangle $ADE$ is a right triangle.
2019 Saudi Arabia JBMO TST 4.4
In the triangle $ABC$, where $<$ $ACB$ $=$ $45$, $O$ and $H$ are the center the circumscribed circle, respectively, the orthocenter. The line that passes through $O$ and is perpendicular to $CO$ intersects $AC$ and $BC$ in $K$, respectively $L$. Show that the perimeter of $KLH$ is equal to the diameter of the circumscribed circle of triangle $ABC$.
2019 Saudi Arabia JBMO Training Test 1.2
Let $AA_1$ and $BB_1$ be heights in acute triangle intersects at $H$. Let $A_1A_2$ and $B_1B_2$ be heights in triangles $HBA_1$ and $HB_1A$, respe. Prove that $A_2B_2$ and $AB$ are parralel.
2019 Saudi Arabia JBMO Training Test 2.4
Let ABCD be a cyclic quadrilateral in which AB = BC and AD =CD. Point M is on the small arc CD of the circle circumscribed to the quadrilateral. The lines BM and CD intersect at point P, and the lines AM and BD intersect at point Q. Prove that PQ is parralel to AC.
2019 Saudi Arabia JBMO Training Test 3.2
An acute triangle ABC is inscribed in a circle C. Tangents in A and C to circle C intersect at F. Segment bisector of AB intersects the line BC at E. Show that the lines FE and AB are parallel.
2019 Saudi Arabia JBMO Training Test 5.2
Two circles, having their centers in A and B, intersect at points M and N. The radii AP and BQ are parallel and are in different semi-planes determined of the line AB. If the external common tangent intersect AB in D, and PQ intersects AB at C, prove that the <CND is right.
2019 Saudi Arabia JBMO Training Test 6.2
The quadrilateral ABCD is circumscribed by a circle C and K, L, M, N are the tangent points of C with the sides AB, BC, CD, DA. Let S be the point of intersection of the lines KM and LN. If the SKBL quadrilateral is cyclic, prove that the quadrilateral SNDM is also cyclic.
Let $ABC$ be a triangle inscribed in the circle $K_1$ and $I$ be center of the inscribed in $ABC$ circle. The lines $IB$ and $IC$ intersect circle $K_1$ again in $J$ and $L$. Circle $K_2$, circumscribed to $IBC$, intersects again $CA$ and $AB$ in $E$ and $F$. Show that $EL$ and $FJ$ intersects on the circle $K_2$.
2018 Saudi Arabia JBMO TST 2.4
Let $ABC$ be a acute triangle in which $O$ and $H$ are the center of the circumscribed circle, respectively the orthocenter. Let $M$ be a point on the small arc $BC$ of the circumscribed circle (different from $B$ and $C$) and be $D, E, F$ be the symmetrical of the point $M$ to the lines $OA, OB, OC$. We note with $K$ the intersection of $BF$ and $CE$ and $I$ is the center of the circle inscribed in the triangle $DEF$.
a) Show that the segment bisectors of the segments $EF$ and $IK$ intersect on the circle
circumscribed to triangle $ABC$.
a) Prove that points $H, K, I$ are collinear.
2018 Saudi Arabia JBMO TST 3.2
Let $ABCD$ be a square inscribed in circle $K$. Let $P$ be a point on the small arc $CD$ of circle $K$. The line $PB$ intersects $AC$ in $E$. The line $PA$ intersects $DB$ in $F$. The circle circumscribed to triangle $PEF$ intersects for second time $K$ in $Q$. Prove that $PQ$ is parallel to $CD$.
Let $AD$ be the perpendicular to the hypotenuse $BC$ of the right triangle $ABC$. Let $DE$ be the height of the triangle $ADB$ and $DZ$ be the height of the triangle $ADC$. On the line $AB$ is chosen the point $N$ so that $CN$ is parallel to $EZ$. Let $A'$ be symmetrical of $A$ to $EZ$ and $I, K$ projections of $A'$ on $AB$, respectively, on $AC$. Prove that $<$ $NA'T$ $=$ $<$ $ADT$, where $T$ is the point of intersection of $IK$ and $DE$.
On the sides $BC$ and $CD$ of the square $ABCD$ of side $1$, are chosen the points $E$, respectively $F$, so that $<$ $EAB$ $=$ $20$ . If $<$ $EAF$ $=$ $45$, calculate the distance from point $A$ to the line $EF$.
2019 Saudi Arabia JBMO TST 3.3
Let $ABC$ be an acute and scalene triangle. Points $D$ and $E$ are in the interior of the triangle so that $\angle DAB = \angle DCB$, $\angle DAC = \angle DBC$, $\angle EAB = \angle EBC$ and $\angle EAC = \angle ECB$. Prove that the triangle $ADE$ is a right triangle.
In the triangle $ABC$, where $<$ $ACB$ $=$ $45$, $O$ and $H$ are the center the circumscribed circle, respectively, the orthocenter. The line that passes through $O$ and is perpendicular to $CO$ intersects $AC$ and $BC$ in $K$, respectively $L$. Show that the perimeter of $KLH$ is equal to the diameter of the circumscribed circle of triangle $ABC$.
Let $AA_1$ and $BB_1$ be heights in acute triangle intersects at $H$. Let $A_1A_2$ and $B_1B_2$ be heights in triangles $HBA_1$ and $HB_1A$, respe. Prove that $A_2B_2$ and $AB$ are parralel.
Let ABCD be a cyclic quadrilateral in which AB = BC and AD =CD. Point M is on the small arc CD of the circle circumscribed to the quadrilateral. The lines BM and CD intersect at point P, and the lines AM and BD intersect at point Q. Prove that PQ is parralel to AC.
2019 Saudi Arabia JBMO Training Test 3.2
An acute triangle ABC is inscribed in a circle C. Tangents in A and C to circle C intersect at F. Segment bisector of AB intersects the line BC at E. Show that the lines FE and AB are parallel.
2019 Saudi Arabia JBMO Training Test 5.2
Two circles, having their centers in A and B, intersect at points M and N. The radii AP and BQ are parallel and are in different semi-planes determined of the line AB. If the external common tangent intersect AB in D, and PQ intersects AB at C, prove that the <CND is right.
The quadrilateral ABCD is circumscribed by a circle C and K, L, M, N are the tangent points of C with the sides AB, BC, CD, DA. Let S be the point of intersection of the lines KM and LN. If the SKBL quadrilateral is cyclic, prove that the quadrilateral SNDM is also cyclic.
2019 Saudi Arabia JBMO Training Test 7.3
Consider a triangle $ABC$ and let $M$ be the midpoint of the side $BC$. Suppose $\angle MAC = \angle ABC$ and $\angle BAM = 105^o$. Find the measure of $\angle ABC$.
Consider a triangle $ABC$ and let $M$ be the midpoint of the side $BC$. Suppose $\angle MAC = \angle ABC$ and $\angle BAM = 105^o$. Find the measure of $\angle ABC$.
2019 Saudi Arabia JBMO Training Test 8.1
Let $E$ be a point lies inside the parallelogram $ABCD$ such that $\angle BCE = \angle BAE$. Prove that the circumcenters of triangles $ABE,BCE,CDE,DAE$ are concyclic.
Let $E$ be a point lies inside the parallelogram $ABCD$ such that $\angle BCE = \angle BAE$. Prove that the circumcenters of triangles $ABE,BCE,CDE,DAE$ are concyclic.
2019 Saudi Arabia JBMO Training Test 9.2
In triangle $ABC$ point $M$ is the midpoint of side $AB$, and point $D$ is the foot of altitude $CD$. Prove that $\angle A = 2\angle B$ if and only if $AC = 2MD$
source: https://pregatirematematicaolimpiadejuniori.wordpress.com/
In triangle $ABC$ point $M$ is the midpoint of side $AB$, and point $D$ is the foot of altitude $CD$. Prove that $\angle A = 2\angle B$ if and only if $AC = 2MD$
In a circle $O$, there are six points, $ A$, $ B$, $C$, $D$, $E$, $F$ in a counterclockwise order such that $BD \perp CF$ , and $CF$, $BE$, $AD$ are concurrent. Let the perpendicular from $B$ to $AC$ be $M$, and the perpendicular from $D$ to $CE$ be $N$. Prove that $AE \parallel MN$.
In a triangle $ABC$, let $K$ be a point on the median $BM$ such that $CM = CK$. It turned out that $\angle CBM = 2\angle ABM$. Show that $BC = KM$.
Let $BB'$, $CC'$ be the altitudes of an acute-angled triangle $ABC$. Two circles passing through $A$ and $C'$ are tangent to $BC$ at points $P$ and $Q$. Prove that $A, B', P, Q$ are concyclic.
Let $BB_1$ and $CC_1$ be the altitudes of acute-angled triangle $ABC$, and $A_0$ is the midpoint of $BC$. Lines $A_0B_1$ and $A_0C_1$ meet the line passing through $A$ and parallel to $BC$ at points $P$ and $Q$. Prove that the incenter of triangle $PA_0Q$ lies on the altitude of triangle $ABC$.
source: https://pregatirematematicaolimpiadejuniori.wordpress.com/
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