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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