Geometry Problems from Junior Balkan MOs

Junior Balkan MO Shortlist 2009 – 2016 with aops links 2009 JBMO Shortlist G1 Parallelogram ABCD is given with AC  BD , and O point of intersection of AC and BD . Circle with center at O and radius OA intersects extensions of AD and AB at points G and L , respectively. Let Z be intersection point of lines BD and GL . Prove that ZCA  90o . 2009 JBMO Shortlist G2 In right trapezoid ABCD  AB CD  the angle at vertex B measures 75o . Point H is the foot of the perpendicular from point A to the line BC . If BH  DC and AD  AH  8 , find the area of ABCD . 2009 JBMO Shortlist G3 Parallelogram ABCD with obtuse angle ABC is given. After rotation of the triangle ACD around the vertex C , we get a triangle CDA , such that points B, C and D are collinear. Extensions of median of triangle CDA that passes through D intersects the straight line BD at point P . Prove that PC is the bisector of the angle BPD . 2009 JBMO Shortlist G4 problem 1 Let ABCDE be a convex pentagon such that AB  CD  BC  DE and k half circle with center on side AE that touches sides AB, BC, CD and DE of pentagon, respectively, at points P, Q, R and S (different from vertices of pentagon). Prove that PS AE . 2009 JBMO Shortlist G5 Let A, B, C and O be four points in plane, such that ABC  90o and OA  OB  OC . Define the point D  AB and the line l such that D  l , AC  DC and l  AO . Line l cuts AC at E and circumcircle of ABC at F . Prove that the circumcircles of triangles BEF and CFD are tangent at F . 2010 JBMO Shortlist G1 Consider a triangle ABC with ACB  90o .Let F be the foot of the altitude from C . Circle  touches the line segment FB at point P, the altitude CF at point Q and the circumcircle of ABC at point R. Prove that points A, Q, R are collinear and AP  AC. 2010 JBMO Shortlist G2 Consider a triangle ABC and let M be the midpoint of the side BC. Suppose MAC  ABC and BAM  105o . Find the measure of ABC. 2010 JBMO Shortlist G3 Let ABC be an acute-angled triangle. A circle 1 (O1 , R1 ) passes through points B and C and meets the sides AB and AC at points D and E , respectively. Let 2 (O2 , R2 ) be the circumcircle of the triangle ADE. . Prove that O1O2 is equal to the circumradius of the triangle ABC.

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Geometry Problems from Junior Balkan MOs

2010 JBMO Shortlist G4 problem 3 Let AL and BK be angle bisectors in the non-isosceles triangle ABC ( L  BC, K  AC ). The perpendicular bisector of BK intersects the line AL at point M . Point N lies on the line BK such that LN MK . Prove that LN  NA . 2011 JBMO Shortlist G1 Let ABC be an isosceles triangle with AB  AC . On the extension of the side CA we consider the point D such that AD  AC . The perpendicular bisector of the segment BD meets the internal and the external bisectors of the angle BAC at the points E and Z , respectively. Prove that the points A, E, D, Z are concyclic. 2011 JBMO Shortlist G2 Let AD, BF and CE be the altitudes of ABC . A line passing through D and parallel to AB intersects the line EF at the point G . If H is the orthocenter of ABC , find the angle CGH . 2011 JBMO Shortlist G3 Let ABC be a triangle in which ( BL is the angle bisector of ABC altitude of ABC

 H  BC 

 L  AC  ,

AH is an

and M is the midpoint of the side AB . It is known that the

midpoints of the segments BL and MH coincides. Determine the internal angles of triangle ABC . 2011 JBMO Shortlist G4 Point D lies on the side BC of ABC . The circumcenters of ADC and BAD are O1 and O2 , respectively and O1O2 AB . The orthocenter of ADC is H and AH  O1O2 . Find the angles of

ABC if 2m  C   3m  B  .

. 2011 JBMO Shortlist G5 Inside the square ABCD , the equilateral triangle ABE is constructed. Let M be an interior point of the triangle ABE such that MB  2, MC  6, MD  5 and ME  3 . Find the area of the square ABCD . 2011 JBMO Shortlist G6 problem 4 Let ABCD be a convex quadrilateral, E and F points on the sides AB and CD , respectively, such that AB : AE  CD : DF  n . Denote by S the area of the quadrilateral AEFD . Prove that AB·CD  n(n  1) AD 2  nDA·BC S . 2n 2 2012 JBMO Shortlist G1 Let ABC be an equlateral triangle and P a point on the circumcircle of the triangle ABC , and distinct from A, B and C . If the lines through P and parallel to BC, CA, AB intersect the lines CA, AB, BC at M , N , Q respectively, prove that M , N and Q are collinear.

Geometry Problems from IMOs blogspot page: Romantics of Geometry facebook group:

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Geometry Problems from Junior Balkan MOs

2012 JBMO Shortlist G2 Let ABC be an isosceles triangle with AB  AC . Let also c  K , KC  be a circle tangent to the line AC at point C which it intersects the segment BC again at an interior point H . Prove that HK  AB . 2012 JBMO Shortlist G3 Let AB and CD be chords in a circle of center O with A, B, C, D distinct, and let the lines AB and CD meet at a right angle at point E . Let also M and N be the midpoints of AC and BD , respectively. If MN  OE , prove that AD BC . 2012 JBMO Shortlist G4 Let ABC be an acute-angled triangle with circumcircle  , and let O, H be the triangle’s circumcenter and orthocenter respectively. Let also A be the point where the angle bisector of angle BAC meets  . If AH  AH , find the measure of the angle BAC . 2012 JBMO Shortlist G5 problem 2 Let the circles k1 and k 2 intersect at two distinct points A and B , and let t t be a common tangent of k1 and k 2 , that touches k1 and k 2 at M and N , respectively. If

t  AM and MN  2 AM , evaluate NMB . 2012 JBMO Shortlist G6 Let O1 be a point in the exterior of the circle c  O, R  and let O1 N , O1D be the tangent segments from O1 to the circle. On the segment O1 N consider the point B such that BN  R . Let the line from B parallel to ON , intersect the segment O1 D at C . If A is a point on the segment O1 D , other than C so that BC  BA  a , and if c  K , r  is the incircle of the triangle O1 AB find the area of ABC in terms of a, R, r . 2012 JBMO Shortlist G7 (ROM) Let MNPQ be a square of side length 1, and A, B, C, D points on the sides MN , NP, PQ and 5 . Can the set  AB, BC, CD, DA be partitioned into two 4 subsets S1 and S 2 of two elements each such that both the sum of the elements of S1 and the sum QM respectively such that AC · BD 

of the elements of S 2 are positive integers? by Flavian Georgescu 2013 JBMO Shortlist G1 (ALB) Let  AB  be a diameter of a circle  and center O , OC a radius of  perpendicular to AB , M be a point of the segment  OC  . Let N be the second intersection point of line AM with

 and P the intersection point of the tangents of  at points N and B. Prove that points M , O, P, N are cocyclic. Geometry Problems from IMOs blogspot page: Romantics of Geometry facebook group:

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Geometry Problems from Junior Balkan MOs

2013 JBMO Shortlist G2 (CYP) Circles 1 , 2 are externally tangent at point M and tangent internally with circle 3 at points K and L respectively. Let A and B be the points that their common tangent at point M of circles 1 and 2 intersect with circle 3 . Prove that if KAB  LAB then the segment  AB 

is diameter of circle 3 . by Theoklitos Paragyiou 2013 JBMO Shortlist G3 problem 2 (FYROM) Let ABC be an acute triangle with AB  AC and O be the center of its circumcircle  . Let D be a point on the line segment BC such that BAD  CAO . Let E be the second point of intersection of  and the line AD . If M , N and P are the midpoints of the line segments  BE  , OD and  AC  respectively, show that the points M , N and P are collinear. by Stefan Lozanovski 2013 JBMO Shortlist G4 (BIH) Let I be the incenter and AB the shortest side of a triangle ABC. The circle with center I and passing through C intersects the ray AB at the point P and the ray BA at the point Q . Let D be the point where the excircle of the triangle ABC belonging to angle A touches the side BC , and let E be the symmetric of the point C with respect to D . Show that the lines PE and CQ are perpendicular. 2013 JBMO Shortlist G5 (BUL) A circle passing through the midpoint M of side BC and the vertex A of a triangle ABC , intersects sides  AB  and  AC  for the second time at points P and Q, respectively. Prove that 1 if BAC  60 then AP  AQ  PQ  AB  AC  BC. 2

2013 JBMO Shortlist G6 (CYP) Let P and Q be the midpoints of the sides BC and CD , respectively in a rectangle ABCD . Let K

and M be the intersections of the line PD with the lines QB and QA respectively, and let

N be the intersection of the lines PA and QB. . Let X , Y and segments ,

AN , KN and

AM

Z be the midpoints of the

respectively. Let l1 be the line passing through

X and

perpendicular to MK , l2 be the line passing through Y and perpendicular to AM and l3 the line passing through Z and perpendicular to KN . . Prove that the lines l1 , l2 and l3 are concurrent. by Theoklitos Paragyiou 2014 JBMO Shortlist G1  Let ABC be a triangle with m  B   m  C   40 Line bisector of B intersects AC at point D . Prove that BD  DA  BC . Geometry Problems from IMOs blogspot page: Romantics of Geometry facebook group:

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Geometry Problems from Junior Balkan MOs

2014 JBMO Shortlist G2 (GRE) Acute-angled triangle ABC with AB  AC  BC and let c  O, R  be it’s circumicircle. Diametes BD and CE ar drawn. Circle c1  A, AE  intersects $AC$at K Circle c2  A, AD  intersects BA at L ( A lies between B και L ).Prove that lines EK and DL intersect at circle

.

by Evangelos Psychas 2014 JBMO Shortlist G3 problem 2 Consider an acute triangle ABC with area S. Let CD  AB ( D  AB) , DM  AC (M  AC ) and DN  BC ( N  BC ) . Denote by H1 and H 2 the orthocenters of the triangles MNC and MND

respectively. Find the area of the quadrilateral AH1BH 2 in terms of S. 2014 JBMO Shortlist G4 Let ABC be a triangle such that AB  AC . Let M be the midpoint of , BC , H be the orthocenter of triangle ABC , O1 be the midpoint of AH , O2 the circumcentre of triangle BCH . Prove that O1 AMO2 is a parallelogram. 2014 JBMO Shortlist G5 Let ABC be a triangle with AB  BC and let BD be the internal bisector of ABC, ,  D  AC  . Denote by M the midpoint of the arc AC which contains point B: The circumscribed circle of the triangle BDM intersects the segment AB at point K  B . Let J be the reflection of A with respect to K . If DJ  AM  O , prove that the points J , B, M , O belong to the same circle. 2014 JBMO Shortlist G6 (ROM) Let ABCD be a quadrilateral whose diagonals are not perpendicular and whose sides AB and CD are not parallel. Let O be the intersection of its diagonals. Denote with H1 and H 2 the orthocenters of triangles OAB and OCD , respectively. If M and N are the midpoints of the segment lines  AB  and CD  , respectively, prove that the lines H1 H 2 and MN are parallel if and only if AC  BD . by Flavian Georgescu 2015 JBMO Shortlist G1 (MNE) Around the triangle ABC the circle is circumscribed, and at the vertex C tangent t to this circle is drawn. The line p , which is parallel to this tangent intersects the lines BC and AC at the points D and E , respectively. Prove that the points A, B, D, E belong to the same circle. 2015 JBMO Shortlist G2 (MOL) The point P is outside the circle  . Two tangent lines, passing from the point P touch the circle  at the points A and B . The median AM  M  BP  intersects the circle  at the point

C and the line PC intersects again the circle  at the point D . Prove that the lines AD and BP are parallel.

Geometry Problems from IMOs blogspot page: Romantics of Geometry facebook group:

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Geometry Problems from Junior Balkan MOs

2015 JBMO Shortlist G3 (GRE) Let c  c  O, R  be a circle with center O and radius R and A, B be two points on it, not belonging to the same diameter. The bisector of angle ABO intersects the circle c at point C , the circumcircle of the triangle AOB , say c1 at point K and the circumcircle of the triangle AOC , say c2 at point L . Prove that point K is the circumcircle of the triangle AOC and that point L is the incenter of the triangle AOB . by Evangelos Psychas 2015 JBMO Shortlist G4 problem 3 (CYP) Let ABC be an acute triangle. Lines l1 , l2 are perpendicular to AB at the points A and B , respectively. The perpendicular lines from the midpoints M of AB to the sides of the triangle AC , BC intersect l1 , l2 at the points E , F , respectively. If D είναι ηο ζημείο ηομής ηων EF και MC , να αποδείξεηε όηι ADB  EMF . by Theoklitos Paragyiou 2015 JBMO Shortlist G5 (ROM) Let ABC be an acute triangle with AB  AC . The incircle  of the triangle κύκλος touches the sides BC, CA and AB at D, E and F , respectively. The perpendicular line erected at C onto BC meets EF at M , and similarly the perpendicular line erected at B onto BC meets EF at N . The line DM meets  again in P , and the line DN meets  again at Q . Prove that DP  DQ . by Ruben Dario and Leo Giugiuc 2016 JBMO Shortlist G1 (GRE) Let ABC be an acute angled triangle, let O be its circumcentre, and let D, E, F be points on the sides BC, CA, AB , respectively. The circle (c1 ) of radius FA , centred at F , crosses the segment OA at A and the circumcircle (c) of the triangle ABC again at K . Similarly, the circle (c2 ) of radius DB , centred at D , crosses the segment  OB  at B and the circle (c) again at L . Finally, the circle (c3 ) of radius EC , centred at E , crosses the segment  OC  at C  and the circle (c) again at M . Prove that the quadrilaterals BKFA, CLDB and AMEC are all cyclic, and their circumcircles share a common point. by Evangelos Psychas 2016 JBMO Shortlist G2  Let ABC be a triangle with BAC  60 . Let D and E be the feet of the perpendiculars from A to the external angle bisectors of ABC and ACB , respectively. Let O be the circumcenter of the triangle ABC . Prove that the circumcircles of the triangles ADE and BOC are tangent to each other. 2016 JBMO Shortlist G3 problem 1 (BUL) A trapezoid ABCD ( AB CD , AB  CD ) is circumscribed. The incircle of triangle ABC touches the lines AB and AC at M and N , respectively. Prove that the incenter of the trapezoid lies on the line MN . Geometry Problems from IMOs blogspot page: Romantics of Geometry facebook group:

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Geometry Problems from Junior Balkan MOs

2016 JBMO Shortlist G4 Let ABC be an acute angled triangle whose shortest side is BC . Consider a variable point P on the side BC , and let D and E be points on AB and AC , respectively, such that BD  BP and CP  CE . Prove that, as P traces BC , the circumcircle of the triangle ADE passes through a fixed point. 2016 JBMO Shortlist G5 Let ABC be an acute angled triangle with orthocenter H and circumcenter O . Assume the circumcenter X of BHC lies on the circumcircle of ABC . Reflect O across X to obtain O , and let the lines XH and OA meet at K . Let L, M and N be the midpoints of  XB  ,  XC  and

 BC  , respectively. Prove that the points

K , L, M and K , L, M , N are cocyclic.

2016 JBMO Shortlist G6 (BUL) Given an acute triangle ABC , erect triangles ABD and ACE externally, so that ADB  AEC  90o and BAD  CAE . Let A1  BC, B1  AC and C1  AB be the feet of the altitudes of the triangle ABC , and let K and K , L be the midpoints of [ BC1 ] and BC1 , CB1 , respectively. Prove that the circumcenters of the triangles AKL, A1B1C1 and DEA1 are collinear. 2016 JBMO Shortlist G7 (CYP) Let AB be a chord of a circle (c) centered at O , and let K be a point on the segment AB such that AK  BK . Two circles through K , internally tangent to (c) at A and B , respectively, meet again at L . Let P be one of the points of intersection of the line KL and the circle (c) , and let the lines AB and LO meet at M . Prove that the line MP is tangent to the circle (c) . by Theoklitos Paragyiou 2017 JBMO Problem 3 Let ABC be an acute triangle such that AB  AC , with circumcircle  and circumcenter O . Let M be the midpoint of BC and D be a point on  such that AD  BC . Let T be a point such that BDCT is a parallelogram and Q a point on the same side of BC as A such that BQM  BCA and CQM  CBA . Let the line AO intersect  at E , ( E  A ) and let the circumcircle of ETQ intersect  at point X  E . Prove that the points A, M and X are collinear.

Geometry Problems from IMOs blogspot page: Romantics of Geometry facebook group:

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