geometry problems from Kharkiv City Olympiads (from Ukraine) [grades \ge 8]
with aops links in the names
2013 Kharkiv City MO 8.3
In the isosceles triangle ABC on the base BC, point M is marked so that the segment MS is equal to the height of the triangle ABC drawn to this side. Point K is marked on the side AB so that the angle KMC is right . Find the angle ACK. Justify the answer.
2013 Kharkiv City MO 9.4
On the side AB of the triangle ABC, the point D is chosen such that AC = CD. On the arc BC of the circumscribed circle of the triangle BCD (not containing the point D), the point E is selected such that \angle ACB = \angle ABE. On the extension of the segment BC beyond point C, a point F is marked such that CE = CF. Prove that AB = AF
source: https://mathedu.kharkiv.ua/examples/sef5/
with aops links in the names
collected inside aops :
it didn't take place in 2020
2012 - 2021
In triangle ABC, \angle A=120^o, \angle B= 40^o. Let AD be the angle bisector of \angle A, and point E on the side of the AC be such that CE = BD. Prove that AD \perp BE.
2012 Kharkiv City MO 9.4
In triangle ABC, points P and Q are the intersection points of a straight line parallel to BC , passing through the vertex A, with bisectors of the external angles B and C of the triangle, respectively. The perpendicular on the line BP at point P, and the perpendicular on the line CQ at point Q, intersect at the point R. Let I be the incenter of the triangle ABC. Prove that AI=AR.
In triangle ABC, points P and Q are the intersection points of a straight line parallel to BC , passing through the vertex A, with bisectors of the external angles B and C of the triangle, respectively. The perpendicular on the line BP at point P, and the perpendicular on the line CQ at point Q, intersect at the point R. Let I be the incenter of the triangle ABC. Prove that AI=AR.
2012 Kharkiv City MO 10.4
In the acute-angled triangle ABC on the sides AC and BC, points D and E are chosen such that points A, B, E, and D lie on one circle. The circumcircle of triangle DEC intersects side AB at points X and Y. Prove that the midpoint of segment XY is the foot of the altitude of the triangle, drawn from point C.
In the acute-angled triangle ABC on the sides AC and BC, points D and E are chosen such that points A, B, E, and D lie on one circle. The circumcircle of triangle DEC intersects side AB at points X and Y. Prove that the midpoint of segment XY is the foot of the altitude of the triangle, drawn from point C.
2012 Kharkiv City MO 11.4
The incircle \omega of triangle ABC touches its sides BC, CA and AB at points D, E and E, respectively. Point G lies on circle \omega in such a way that FG is a diameter. Lines EG and FD intersect at point H. Prove that AB \parallel CH.
The incircle \omega of triangle ABC touches its sides BC, CA and AB at points D, E and E, respectively. Point G lies on circle \omega in such a way that FG is a diameter. Lines EG and FD intersect at point H. Prove that AB \parallel CH.
2013 Kharkiv City MO 8.3
In the isosceles triangle ABC on the base BC, point M is marked so that the segment MS is equal to the height of the triangle ABC drawn to this side. Point K is marked on the side AB so that the angle KMC is right . Find the angle ACK. Justify the answer.
2013 Kharkiv City MO 9.4
On the side AB of the triangle ABC, the point D is chosen such that AC = CD. On the arc BC of the circumscribed circle of the triangle BCD (not containing the point D), the point E is selected such that \angle ACB = \angle ABE. On the extension of the segment BC beyond point C, a point F is marked such that CE = CF. Prove that AB = AF
2013 Kharkiv City MO 10.4
The pentagon ABCDE is inscribed in the circle \omega. Let T be the intersection point of the diagonals BE and AD. A line is drawn through the point T parallel to CD, which intersects AB and CE at points X and Y, respectively. Prove that the circumscribed circle of the triangle AXY is tangent to \omega.
The pentagon ABCDE is inscribed in the circle \omega. Let T be the intersection point of the diagonals BE and AD. A line is drawn through the point T parallel to CD, which intersects AB and CE at points X and Y, respectively. Prove that the circumscribed circle of the triangle AXY is tangent to \omega.
2013 Kharkiv City MO 11.4
In the triangle ABC, the heights AA_1 and BB_1 are drawn. On the side AB, points M and K are chosen so that B_1K\parallel BC and A_1 M\parallel AC. Prove that the angle AA_1K is equal to the angle BB_1M.
In the triangle ABC, the heights AA_1 and BB_1 are drawn. On the side AB, points M and K are chosen so that B_1K\parallel BC and A_1 M\parallel AC. Prove that the angle AA_1K is equal to the angle BB_1M.
2014 Kharkiv City MO 8.5
In the right triangle ABC with hypotenuse AB, points D and E are selected on side AC and points F and G are selected on side AC such that \angle ABD = \angle DBE= \angle EBC and \angle BAF = \angle FAG = \angle GAC. Segments AF and BD intersect at point K. Prove that the triangle EGK is equilateral.
In the right triangle ABC with hypotenuse AB, points D and E are selected on side AC and points F and G are selected on side AC such that \angle ABD = \angle DBE= \angle EBC and \angle BAF = \angle FAG = \angle GAC. Segments AF and BD intersect at point K. Prove that the triangle EGK is equilateral.
2014 Kharkiv City MO 9.5
In the triangle ABC with AB <AC is satisfied, let point I be the center of the inscribed circle of this triangle. Point E is selected on the side AC such that AE = AB. Point G lies on the straight line EI such that \angle IBG = \angle CBA, and points E and C lie on opposite sides of I. Prove that the line AI, the perpendicular on AE at point E, and the bisector of angle \angle BGI intersect at one point.
In the triangle ABC with AB <AC is satisfied, let point I be the center of the inscribed circle of this triangle. Point E is selected on the side AC such that AE = AB. Point G lies on the straight line EI such that \angle IBG = \angle CBA, and points E and C lie on opposite sides of I. Prove that the line AI, the perpendicular on AE at point E, and the bisector of angle \angle BGI intersect at one point.
2014 Kharkiv City MO 10.4
Let ABCD be a square. The points N and P are chosen on the sides AB and AD respectively, such that NP=NC. The point Q on the segment AN is such that that \angle QPN=\angle NCB. Prove that \angle BCQ=\frac{1}{2}\angle AQP.
Let ABCD be a square. The points N and P are chosen on the sides AB and AD respectively, such that NP=NC. The point Q on the segment AN is such that that \angle QPN=\angle NCB. Prove that \angle BCQ=\frac{1}{2}\angle AQP.
In the convex quadrilateral of the ABCD, the diagonals of AC and BD are mutually perpendicular and intersect at point E. On the side of AD, a point P is chosen such that PE = EC. The circumscribed circle of the triangle BCD intersects the segment AD at the point Q. The circle passing through point A and tangent to the line EP at point P intersects the segment AC at point R. It turns out that points B, Q, R are collinear. Prove that \angle BCD = 90^o.
2015 Kharkiv City MO 8.5
The diagonals AC and BD of a convex quadrilateral intersect at a point O. Points M and .N are chosen on the segments AC and BD respectively such that AM = OC and DN = BO. Prove that if AB \nparallel CD, then it is possible to form a triangle from segments AB, CD, and MN.
The diagonals AC and BD of a convex quadrilateral intersect at a point O. Points M and .N are chosen on the segments AC and BD respectively such that AM = OC and DN = BO. Prove that if AB \nparallel CD, then it is possible to form a triangle from segments AB, CD, and MN.
2015 Kharkiv City MO 9.5
The circles \omega_1 and \omega_2 intersect at points P and Q, are inside the circle \omega and tangent to it at points A_2 and A_2, respectively. The line PQ intersects the circle \omega at points B and D. Let E_1 and F_1 be the second intersection points of the lines A_1B and A_1D with \omega_1; Let E_1 and E_2 be the second intersection points of A_2B and A_2D with \omega_2. Prove that the points E_1,E_2,F_1 and F_2 lie on the same circle.
The circles \omega_1 and \omega_2 intersect at points P and Q, are inside the circle \omega and tangent to it at points A_2 and A_2, respectively. The line PQ intersects the circle \omega at points B and D. Let E_1 and F_1 be the second intersection points of the lines A_1B and A_1D with \omega_1; Let E_1 and E_2 be the second intersection points of A_2B and A_2D with \omega_2. Prove that the points E_1,E_2,F_1 and F_2 lie on the same circle.
2015 Kharkiv City MO 10.3
On side AB of triangle ABC, point M is selected. A straight line passing through M intersects the segment AC at point N and the ray CB at point K. The circumscribed circle of the triangle AMN intersects \omega, the circumscribed circle of the triangle ABC, at points A and S. Straight lines SM and SK intersect with \omega for the second time at points P and Q, respectively. Prove that AC = PQ.
On side AB of triangle ABC, point M is selected. A straight line passing through M intersects the segment AC at point N and the ray CB at point K. The circumscribed circle of the triangle AMN intersects \omega, the circumscribed circle of the triangle ABC, at points A and S. Straight lines SM and SK intersect with \omega for the second time at points P and Q, respectively. Prove that AC = PQ.
2015 Kharkiv City MO 11.3
In the rectangle ABCD, point M is the midpoint of the side BC. The points P and Q lie on the diagonal AC such that \angle DPC = \angle DQM = 90^o. Prove that Q is the midpoint of the segment AP.
In the rectangle ABCD, point M is the midpoint of the side BC. The points P and Q lie on the diagonal AC such that \angle DPC = \angle DQM = 90^o. Prove that Q is the midpoint of the segment AP.
2016 Kharkiv City MO 8.3
On the side BC of the triangle ABC, point M is selected, on the side AC point N and on the side AB point K, such that BM = BK and CM = CN. The perpendicular dropped on the segment MK from point B intersects the perpendicular dropped on the segment MN from point C, at point I. Prove that \angle IKA= \angle INC
On the side BC of the triangle ABC, point M is selected, on the side AC point N and on the side AB point K, such that BM = BK and CM = CN. The perpendicular dropped on the segment MK from point B intersects the perpendicular dropped on the segment MN from point C, at point I. Prove that \angle IKA= \angle INC
2016 Kharkiv City MO 9.3
An isosceles triangle ABC with the base BC is given. The circle \omega it tangent to the line AC at point C and intersects the ray AB at points X and Y. Prove that \angle BCX = \angle BCY.
An isosceles triangle ABC with the base BC is given. The circle \omega it tangent to the line AC at point C and intersects the ray AB at points X and Y. Prove that \angle BCX = \angle BCY.
2016 Kharkiv City MO 10.3
Let AD be the bisector of an acute-angled triangle ABC. The circle circumscribed around the triangle ABD intersects the straight line perpendicular to AD that passes through point B, at point E. Point O is the center of the circumscribed circle of triangle ABC. Prove that the points A, O, E lie on the same line.
Let AD be the bisector of an acute-angled triangle ABC. The circle circumscribed around the triangle ABD intersects the straight line perpendicular to AD that passes through point B, at point E. Point O is the center of the circumscribed circle of triangle ABC. Prove that the points A, O, E lie on the same line.
The circle \omega passes through the vertices B and C of triangle ABC and intersects its sides AC,AB at points A,E, respectively. On the ray BD, a point K such that BK = AC is chosen , and on the ray CE, a point L such that CL = AB is chosen. Prove that the center O of the circumscribed circle of the triangle AKL lies on the circle \omega.
2017 Kharkiv City MO 6.3 7.3
The square is divided into 5 rectangles as shown. It is known that all rectangles of a partition have the same area and that the length of the segment AB is 5. Find the length of the segment CD. Justify the answer.
The square is divided into 5 rectangles as shown. It is known that all rectangles of a partition have the same area and that the length of the segment AB is 5. Find the length of the segment CD. Justify the answer.
2017 Kharkiv City MO 8.3
Let ABCD be a convex quadrangle with \angle A = \angle B = 60^o and \angle CAB = \angle CBD. Prove that AC = BD.
Let ABCD be a convex quadrangle with \angle A = \angle B = 60^o and \angle CAB = \angle CBD. Prove that AC = BD.
2017 Kharkiv City MO 9.3
A circle intersects the sides AB, BC, CA of the triangle ABC at points R and S, M and N, P and Q, respectively, so that point S lies on the segment RB, point N on the segment MC, point Q on the segment PA. Prove that if RS=MN=PQ and QR=SM=NP, then triangle ABC is equilateral.
2017 Kharkiv City MO 10.4
In the quadrangle ABCD, the angle at the vertex A is right. Point M is the midpoint of the side BC. It turned out that \angle ADC = \angle BAM. Prove that \angle ADB = \angle CAM.
A circle intersects the sides AB, BC, CA of the triangle ABC at points R and S, M and N, P and Q, respectively, so that point S lies on the segment RB, point N on the segment MC, point Q on the segment PA. Prove that if RS=MN=PQ and QR=SM=NP, then triangle ABC is equilateral.
In the quadrangle ABCD, the angle at the vertex A is right. Point M is the midpoint of the side BC. It turned out that \angle ADC = \angle BAM. Prove that \angle ADB = \angle CAM.
The quadrilateral ABCD is inscribed in the circle \omega. Lines AD and BC intersect at point E. Points M and N are selected on segments AD and BC, respectively, so that AM: MD = BN: NC. The circumscribed circle of the triangle EMN intersects the circle \omega at points X and Y. Prove that the lines AB, CD and XY intersect at the same point or are parallel.
2018 Kharkiv City MO 8.5
Let ABCD be a convex quadrilateral with \angle A = 45^o, \angle ADC = \angle ACD = 75^o and AB = CD = 1. Find the length of the segment BC. Justify the answer.
Let ABCD be a convex quadrilateral with \angle A = 45^o, \angle ADC = \angle ACD = 75^o and AB = CD = 1. Find the length of the segment BC. Justify the answer.
2018 Kharkiv City MO 9.2
On the base AC of the isosceles triangle ABC, the point D is selected so that CD = 2AD. On the extension of the segment BD beyond point D, point E is selected such that BD = DE. Prove that AE =DE.
On the base AC of the isosceles triangle ABC, the point D is selected so that CD = 2AD. On the extension of the segment BD beyond point D, point E is selected such that BD = DE. Prove that AE =DE.
On the sides AB, AC ,BC of the triangle ABC, the points M, N, K are selected, respectively, such that AM = AN and BM = BK. The circle circumscribed around the triangle MNK intersects the segments AB and BC for the second time at points P and Q, respectively. Lines MN and PQ intersect at point T. Prove that the line CT bisects the segment MP.
2018 Kharkiv City MO 11.4
The line \ell parallel to the side BC of the triangle ABC, intersects its sides AB,AC at the points D,E, respectively. The circumscribed circle of triangle ABC intersects line \ell at points F and G, such that points F,D,E,G lie on line \ell in this order. The circumscribed circles of the triangles FEB and DGC intersect at points P and Q. Prove that points A, P and Q are collinear.
The line \ell parallel to the side BC of the triangle ABC, intersects its sides AB,AC at the points D,E, respectively. The circumscribed circle of triangle ABC intersects line \ell at points F and G, such that points F,D,E,G lie on line \ell in this order. The circumscribed circles of the triangles FEB and DGC intersect at points P and Q. Prove that points A, P and Q are collinear.
2019 Kharkiv City MO 8.4
Let ABCD be parallelogram with \angle ADC = 40^o. Let K be a point such that the segment AK intersects the side BC, AK = BC and \angle BAK = 80^o. Let L be a point such that the segment CL intersects the side AD, CL = AB and \angle BCL = 80^o. Find the angles of the triangle BKL. Justify your answer.
Let ABCD be parallelogram with \angle ADC = 40^o. Let K be a point such that the segment AK intersects the side BC, AK = BC and \angle BAK = 80^o. Let L be a point such that the segment CL intersects the side AD, CL = AB and \angle BCL = 80^o. Find the angles of the triangle BKL. Justify your answer.
2019 Kharkiv City MO 9.4
Two circles \gamma and \omega are such that the circle \omega passes through the center of the circle \gamma. Let A and B be the intersection points of \gamma and \omega . On the circle \omega, a random point P, is chosen. The lines PA and PB intersect \gamma for the second time at points E and F, respectively. Prove that AB = EF.
Two circles \gamma and \omega are such that the circle \omega passes through the center of the circle \gamma. Let A and B be the intersection points of \gamma and \omega . On the circle \omega, a random point P, is chosen. The lines PA and PB intersect \gamma for the second time at points E and F, respectively. Prove that AB = EF.
2019 Kharkiv City MO 10.5
In triangle ABC, point I is incenter , I_a is the A-excenter. Let K be the intersection point of the BC with the external bisector of the angle BAC, and E be the midpoint of the arc BAC of the circumcircle of triangle ABC. Prove that K is the orthocenter of triangle II_aE.
In triangle ABC, point I is incenter , I_a is the A-excenter. Let K be the intersection point of the BC with the external bisector of the angle BAC, and E be the midpoint of the arc BAC of the circumcircle of triangle ABC. Prove that K is the orthocenter of triangle II_aE.
2019 Kharkiv City MO 11.6
In the acute-angled triangle ABC, let CD, BE be the altitudes. Points F and G are the projections of A and C on the line DE, respectively, H and K are the projections of D and E on the line AC, respectively. The lines HF and KG intersect at point P. Prove that line BP bisects the segment DE.
In the acute-angled triangle ABC, let CD, BE be the altitudes. Points F and G are the projections of A and C on the line DE, respectively, H and K are the projections of D and E on the line AC, respectively. The lines HF and KG intersect at point P. Prove that line BP bisects the segment DE.
In the triangle ABC, on the median BM lies point K such that CK = CM. It is known that \angle CBM = 2\angle ABM. Prove that BC = MK
On the side AB of the triangle ABC, the point K is marked so that AB = CK. The points N and M are the midpoints of the segments AK and BC, respectively. The segments NM and CK intersect at R. Prove that KN=KR.
The inscribed circle \Omega of triangle ABC touches the sides AB and AC at points K and L, respectively. The line BL intersects the circle \Omega for the second time at the point M. The circle \omega passes through the point M and is tangent to the lines AB and BC at the points P and Q, respectively. Let N be the second intersection point of circles \omega and \Omega, which is different from M. Prove that if KM \parallel AC then the points P, N and L lie on one line.
In the triangle ABC, the segment CL is the angle bisector. The C-exscribed circle with center at the point I_c touches the side of the AB at the point D and the extension of sides CA and CB at points P and Q, respectively. It turned out that the length of the segment CD is equal to the radius of this exscribed circle. Prove that the line PQ bisects the segment I_CL.
source: https://mathedu.kharkiv.ua/examples/sef5/
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