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In the World of Mathematics

geometry problems with aops links from 

U sviti matematyky  /   In the World of Mathematics

                                                       Ukrainian Magazine Magazine 


Here I am gonna post all the (Euclidean) geometry problems, from this magazine with aops links in their problem number after I propose them there,

 geometry problems withs aops links
collected inside aops in two parts: (part I) (part II)

                                          all the geometry problems have been posted  below
aops links are under construction

where there is a red colour below, 
a typo has been corrected thanks to Albrecht Hess from Madrid
who started solving them in aops


P26 On the hypotenuse AB of the triangle ABC with \angle C = 90^o and the area S, as on the diameter, was drawn a circle. The points K and M was chosen on the arcs AB and AC correspondingly in such a way that the chord KM is a diameter of a circle. Let P and Q be the bases of the perpendiculars, that are drown from the points A and C on the chords CM and AM correspondingly. Prove, that the area of KPMQ equals S.
(I. Nagel, Eupatoria)

P41 Let O be a center of a circle, circumscribed over \triangle ABC. Perpendicular, drown from the point A on the line CO, cross the line CB in the point M. Find |CM| , if |CA| = a and |CB| = b.

(V. Yasinskyy, Vinnytza)

P48 Points A, B, C and D are chosen on a circle in such a way that the point S of the intersection of the lines AB and CD lies outside the circle and the point T of intersection of the lines AC and BD lies inside the circle. Let points M and N lie on chords BD and AC respectively and K denotes point of intersection of the lines ST and MN. Prove that \frac{MK}{KN} = \frac{TM}{TN }\cdot \frac{BD}{AC} 
(V. Petechuk, Uzhgorod)

P51 All angles of a triangle ABC are less than 90^o, moreover, angle B equals 60^o. Let AM and CK be the altitudes of ABC and P and L be the middle points of AB and CB correspondingly. Prove that the line which pass through B and the point of intersection of PL and KM is the bisector of the angle B.


(I. Nagel, Herson)

P53 Two secants to the circle \omega pass through the point S lying outside \omega. Let A and B, C and D be the intersection points with SA < SB and SC < SD. Denote by T the point of intersection of the chords AD and BC. Prove, that the intersection point of two tangent lines passing through B and D belongs to ST.

(V. Petechuk, Uzhgorod)

P55 Find the polyhedron with 8 triangular faces and maximal volume which is drawn in the fixed sphere.


(O. Kukush, Kyiv)

P59 A circle inscribed into triangle ABC touches side BC in a point E. Segment CD is perpendicular to BC and has the same length as CA. Find a radius of a circle inscribed into triangle BCD if CE = 1 cm., and the length of BD is 2 cm  shorter than the length of BA.


(O. Kukush, Kyiv)

P61 Given a triangle ABC. The perpendiculars to the plane ABC pass through the vertices of the triangle. Points A_1, B_1, C_1 were fi xed on the corresponding perpendiculars at the following way: all of them lie at the same side with respect to ABC. Moreover, the lengths of AA_1, BB_1 and CC_1 equal to the lengths of the corresponding altitudes of ABC. Let S be an intersection point of plains AB_1C_1, A_1BC_1 and A_1B_1C. Find the area of the surface of pyramid SABC.
(V. Yasinskyy, Vinnytsa)

P68 Construct a convex quadrangle if known are the orthogonal projection of the cross point of it diagonals on all four sides.

(V. Yasinskyy, Vinnytsa)

P73 Trapezium ABCD is inscribed into a circle of radius R and circumscribed over a circle of radius r. Find the distance between the centers of these circles.

(R. Ushakov, Kyiv)

P79 A circle \omega is outscribed over an acute triangle ABC. AN and CK are altitudes of ABC. The median BM crosses the circle  \omega in the point P. The point Q is chosen on the section BM such that MQ = MP. Prove that the points B, K, Q are N belong to the same circle.
(I.Nagel, Evpatoriya)

P96 A tetrahedron ABCD is circumscribed around a sphere \omega of the radius r, tangent to the faces ABC, BCD, CDA, DAB in the points D_1, A_1, B_1, C_1 respectively. The lines AA_1, BB_1, CC_1, DD_1 intersect the sphere \omega for the second time at the points A_2, B_2, C_2, D_2 respectively. Prove the inequality
AA_1  \cdot A_1A_2 +   BB_1  \cdot B_1B_2+ CC_1  \cdot  C_1C_2 + DD_1 \cdot D_1D_2 \ge  32r^2

(V. Yasinskyj, Vinnytsa)

P97 Let BM and CN be the bisectors in the triangle ABC (\angle A = 60^o), intersecting in the point I. Let P and Q be the tangency points of the inscribed circle to the sides AB and AC respectively. Denote by O the midpoint of the segment NM. Prove that the points P,O and Q are collinear.
(I. Nagel, Evpatoriya)

P103 Let O be the intersection point of the diagonals in an inscribed quadrilateral ABCD. The points P and Q belong to the rays OA and OB respectively and \angle DAQ = \angle CBP. Prove that the point O and the midpoints of the segments PQ and CD are collinear.
(V. Yasinskyi, Vinnytsa)

P104 The angles of the triangle ABC are less than 120^o. The point O inside the triangle is such that \angle AOB =  \angle BOC =  \angle COA = 120^o. Let M_1,M_2,M_3 be the intersection points of the medians and let H_1,H_2,H_3 be the intersection points of the altitudes in the triangles AOB,BOC,COA respectively. Prove the equality \overrightarrow{M_1H_1} + \overrightarrow{M_2H_2}+\overrightarrow{M_2H_3} = -2\overrightarrow{OM}, where M is the intersection point of the medians in the triangle ABC.


(M. Kurylo, Lypova Dolyna)

P109 Two circles with different radii are tangent to a line \ell in points A and B and intersect one another in points C and D. Let H_1 be the intersection point of the altitudes of the triangle ABC, and let H_2 be the intersection point of the altitudes of the triangle ABD. Prove that H_1CH_2D is a parallelogram.

(V. Yasinsky, Vinnytsa)

P112 Let for a tetrahedron DABC the equalities \frac{DA}{sin \alpha }=\frac{ DB}{sin \beta} = \frac{DC}{sin \gamma} hold, where \alpha, \beta,\gamma are the interfacial angles by the edges DA, DB and DC respectively. Prove that the center of the inscribed sphere, the intersection point of the medians of the triangle ABC and the point D are collinear.

(M. Kurylo, Lypova Dolyna, Sumska obl.)

P114 A point H belongs to the diagonal AC of a convex quadrilateral ABCD and is such, that BH \perp AC. Prove that AB = AD if AB \perp  BC and AO \perp  DH, where O is the center of the circle circumscribed around the triangle ACD.
(V. Yasinskyy, Vinnytsa)

P116 A sphere with center in a point I is inscribed in a trihedral angle OABC. Prove that the planes AOI and BOI are perpendicular if \angle BOC+ \angle AOC =180^o  +\angle  AOB.


(M. Kurylo, Lypova Dolyna, Sumska obl.)

P118, P 126 Construct with help of a compass and a ruler a triangle ABC knowing the vertex A, the midpoint of the side BC and the intersection point of the altitudes.
(V. Yasinskyy, Vinnytsa)

P122 Every diagonal of a convex quadrilateral is a bisector of an angle and a trisector of the opposite one. Find the angles of the quadrilateral
(V. Yasinsky, Vinnytsia)

P128 A triangle ABC is given. A point R belongs to the line AC and C is between the points A and R. The point R belongs to a strait line intersecting the side AB in a point C_1 and the side BC in a point A_1. Let P be the midpoint of the side AC, and let Q be the midpoint of the side A_1C_1. Prove that three cirles, circumsribed around the triangles ABC,A_1BC_1 and PQR respectively are concurrent.

(V. Yasinsky, Vinnytsia)

P133 Inside a convex quadrangle, ABCD, a point M is chosen in an arbitrary way. Four perpendiculars have been drawn from M to the lines containing the sides of the quadrangle: MN \perp AB, MI \perp BC, MH \perp CD and MK \perp DA. Prove, that the doubled size of the quadrangle NIHK is not greater than MA \cdot MC +MB \cdot MD.
(I. Nagel, Evparorija)

P134 Circles \omega_1, \omega_2 and \omega_3 touch the circle \omega in an inner way in points A_1, A_2 and A_3 correspondingly. It is also known that the circles \omega_1 and \omega_2 touch each other in an outer way in the pont B_3, circles \omega_2 and \omega_3 touch each other in an outer way in the pont B_1, and circles \omega_1 and \omega_3 touch each other in an outer way in the pont B_2. Prove that straight lines A_1B_1, A_2B_2 and A_3B_3 have a common point.
(O. Manzjuk, Kyiv)

P139 A circle inscribed in a triangle, ABC, touches the sides AB, BC and AC in points X, Y and Z respectively. The perpendiculars YK \perp AB, XP \perp  AC and ZQ  \perp  BC are constructed. Find the area of XYZ in terms of lengths of XP, Y K and ZQ.
(I. Nagel, Evpatoria)

P140 The points B_1 and C_1 are chosen on the sides AC and AB respectively of an acute triangle, ABC. Let X denote the intersection point of BB_1 and CC_1 and M denote the center of BC. Prove that X is the orthocenter of \triangle ABC provided the quadrangle AB_1XC_1 is inscribed in a circle and B_1M = C_1M.

(V.Duma, A.Prymak, O. Manzjuk, Kyiv)

P143 Let AA_1,BB_1,CC_1 be the bisectors in the  \triangle ABC and let A_2,B_2,C_2 be the tangency points of the incircle to the sides of the triangle. Prove that the area of the triangle \triangle A_2B_2C_2 is not greater than the area of the \triangle A_1B_1C_1.


(R. Ushakov, Kyiv)

P153 The triangles ACB and ADE are oriented in the same way.  We also have that \angle DEA = \angle ACB = 90^o, \angle DAE = \angle BAC, E \ne C.  The line \ell passes through the point D and is perpendicular to the line EC.  Let L be the intersection point of the lines \ell  and AC.  Prove that the points L,E,C,B belong to a common circumference.

(V. Yasinsky, Vinnytsia)

P154  Let ABCD be a trapezoid (BC // AD), denote by E the intersection point of its diagonals and by O the center of the circle circumscribed around the triangle \triangle AED. Let K and L the points on the segments AC and BD respectively such that BK \perp AC and CL \perp BD. Prove that KL \perp OE.
(A. Prymak, Kyiv)

P157 Let A_1,B_1,C_1 be the midpoints of the segments BC,AC,AB of the triangle \triangle ABC respectively. Let H_1,H_2,H_3 be the intersection points of the altitudes of the triangles \triangle AB_1C_1, \triangle BA_1C_1, \triangle CA_1B_1. Prove that the lines A_1H_1,B_1H_2,C_1H_3 are concurrent.
(M. Kurylo, Lypova Dolyna, Sumska obl.)

P168 Let AA_1,BB_1,CC_1 be bisectors in the triangle ABC, let G_1,G_2,G_3 be the intersection points of medians in the triangles AB_1C_1,BA_1C_1 and CA_1B_1 respectively. Prove that the straight lines AG_1,BG_1,CG_1 intersect in a common point.

(M. Kurylo, Lypova Dolyna, Sumska obl.)

P173 Let triangle ABC be inscribed into a circle. Points C and M lie on different arcs of the circle with endpoints A and B. Chords MK and MP intersect AC and BC in the points H and N respectively. Chords AP and BK intersects in the point I. Prove that points H, I and N lies on the same straight line.
(I. Nagel, Evpatoria)

P178 Let ABC be acute-angled triangle, let \omega  be the circle circumscribed around \triangle ABC, let M_1,M_2,M_3 be the midpoints of BC,AB and AC correspondingly. The altitudes from A and C to BC and AB intersect \omega in the points L_1 and L_2. P_3 is the intersection point of the altitudes of the triangle BM_1M_2. Prove that the straight lines M_3P_3 and L_1L_2 are perpendicular.
(O. Chubenko, Pryluky, Chernigivska obl.)

P181 Let M and M_1 be the intersection points of medians in the triangles  \triangle ABC and \triangle A_1B_1C_1,   \angle ACM = \angle A_1C_1M_1 , \angle MBC =  \angle M_1B_1C_1. Is it possible for \triangle ABC and  \triangle A_1B_1C_1 not to be similar?

(V. Duma, Kyiv.)

P186  Let AA_1,BB_1 and CC_1 be the altitudes of the acute triangle ABC.Let AA_2,BB_2,CC_2 be its medians which intersect B_1C_1,A_1C_1 and A_1B_1 in the points A_1,B_3,C_3 correspondingly. Prove that the straight lines A_1A_3,B_1B_3 and C_1C_3 intersect in a common point.
(M. Kurylo, Lypova Dolyna, Sumska obl.)

P187 Construct a triangle ABC if known are the circle \omega, circumscribed around \triangle ABC, a point D on AB, line \ell parallel to AC and the length of BC.
(V. Duma, Kyiv)

P191 (also) Let M be the intersection point of medians in the acute-angled triangle \triangle ABC, \angle BAC = 60^o and \angle BMC = 120^o. Prove that \triangle ABC is equilateral triangle.
(V. Duma, Kyiv)

P193 The circles \omega_1 and \omega_2 with centres O_1,O_2 and radii R_1,R_2 respectively intersect at the points A and B. Tangent lines to \omega_2 and \omega_1 passing through A intersect \omega_1 and \omega_2 in the points C and D respectively. Let E and F be the points on the rays AO_1 and AO_2 such that AE = R_2 and AF = R_1. Let M be the midpoint of EF. Prove that AM \perp CD and CD \le 4AM.

(M. Kurylo, Lypova Dolyna, Sumska obl.)

P201 Triangle ABC is inscribed into the circle \omega. The circle \omega_1 touches the circle \omega in an inner way and touches sides AB and AC in the points M and N. The circle \omega_2 also touches the circle \omega in an inner way and touches sides AB and BC in the points P and K respectively. Prove that NKMP is a parallelogram.

(M. Kurylo, Lypova Dolyna, Sumska obl.)

P204 (also) In the convex pentagon ABCDE  \angle ABC = \angle AED = 90^o and AB  \cdot   ED = BC \cdot  AE. Let F be the intersection point of CE and BD. Prove that AF \perp  BE.

(M. Kurylo, Lypova Dolyna, Sumska obl.)

P206 The diagonals AC and BD of convex quadrilateral ABCD intersect at the point P. The circles circumscribed around \triangle ABP and \triangle  DCP intersect at the point M distinct from P. The circles circumscribed around \triangle BCP and \triangle ADP intersect at the point N distinct from P. Perpendiculars to AC and BD passing through the midpoints of AC and BD intersect at the point O. Prove that the points M,O, P,N lie on the same circle.

(M. Kurylo, Lypova Dolyna, Sumska obl.)

P210 For any point D lying on the side AB of a triangle ABC denote by P and Q the centres of the circles inscribed into \triangle ACD and \triangle  BCD. Find all points D such that the triangle PQD is similar to the triangle ABC.
(B. Rublyov, Kyiv)

P212 Two circles \omega_1 and \omega_2 of different radii intersect at points A and B. The straight line CD touches the circles \omega_1 and \omega_2 at points C and D as well as the straight line EF touches the circles \omega_1 and \omega_2 at points E and F respectively. Let H_1,H_2,H_3,H_4 be the intersection points of the altitudes of triangles EFA,CDA,EFB,CDB. Prove that H_1H_2H_3H_4 is a rectangular.

(M. Kurylo, Lypova Dolyna, Sumska obl.)

P218  (also)Let ABCD be a convex cyclic inscribed quadrilateral. Bisectors of the angles \angle BAD and \angle BCD intersect at the diagonal BD. Let E be the midpoint of BD. Prove that \angle BAE =\angle CAD
(Ì. Kurylo, Lypova Dolyna, Sumska obl.)

P221 Point P is chosen inside the triangle ABC. Denote by X,Y,Z the intersection points of AP,BP,CP with BC,AC,AB respectively. Let M_1,M_2,M_3 be the midpoints of AC,AB,BC and N_1,N_2,N_3 be the midpoints of XZ,XY, YZ respectively. Prove that the straight lines M_1N_1,M_2N_2 and M_3N_3 intersect in a common point.

(O. Chubenko, Pryluky, Chernigivska obl.)

P230 Let \triangle ABC be a triangle such that 3AC = AB+BC. The inscribed circle of \triangle ABC touches the side AC at point K and KL is a diameter of the circle. The straight lines AL and CL intersect BC and AB at A_1 and C_1 respectively. Prove that AC_1 = CA_1.

(A. Gogolev, Kyiv)

P235 [also]The circle \omega inscribed into ABC touches sides BC,AC,AB at K,L,M respectively. The perpendiculars at K,L and M to LM,KM and KL, intersect the circle \omega at P,Q and R respectively. Prove that the straight lines AP,BQ and CR are concurrent.

(Î. Manzjuk, À. Prymak, Kyiv)

P240 The similar isosceles triangles \triangle AC_1B,\triangle BA_1C and \triangle CB_1A with bases AB,BC and AC respectively are constructed externally on the sides of non-isosceles triangle \triangle ABC. Prove that if A_1B_1 = B_1C_1 then \angle BAC_1 = 30^o.

(Å. Tyrkevych, Chernivtsi)

P243 Let I be the incentre of triangle \triangle ABC and r be corresponding inradius. The straight line \ell passing through I intersects the incircle of \triangle ABC at points P and Q and the circumcircle of  \triangle ABC at points M and N, where P lies between M and I. Prove that MP + NQ \ge 2r.

(V. Yasinskyy, Vinnytsya)

P245  Let AC be the longest side of triangle \triangle ABC, BB1 be the altitude and H be the intersection point of the altitudes of triangle \triangle ABC. Prove that if BH = 2B_1H then \triangle ABC is an equilateral triangle.
(Å. Tyrkevych, Chernivtsi)

P247 Let ABCDEF be a regular hexagon. Denote by G, H, I, J, K, L the intersection points of the sides of triangles \triangle ACE and \triangle BDF. Does there exist a bijection f which maps A, B, C, D, E, F onto G, H, I, J, K, L and vice versa such that the images of any four points lying on some straight line belong to some circle?
(O. Kukush, Kyiv)

P249 Let DABC be a regular trihedral pyramid. The points A_1,B_1,C_1 are chosen at lateral edges DA,DB,DC respectively such that the planes ABC and A_1B_1C_1 are parallel. Let O be the circumcenter of DA_1B_1C. Prove that DO is perpendicular to the plane ABC_1.

(M. Kurylo, Lypova Dolyna)

P252 The circles \omega_1 and \omega_2 with centres O_1,O_2 intersect at points A and B. The circle \omega passing through O_1,O_2,A intersects \omega_1 and \omega_2 again in points K,M respectively. Prove that AB is a bisector of \angle KAM or of angle adjacent to \angle KAM.
(T. Tymoshkevych, Kyiv)

P254 The circle \omega passing through the vertices B and C of a triangle \triangle ABC with AB \ne AC intersects the sides AB and AC at R and S. Let M be the midpoint of BC. The straight line perpendicular to MA at A intersects BS and CR at K and T respectively. Prove that if TA = AK then MS = MR.
(O. Klurman, Lviv)

P257 Points M_1 and M_2 lie inside the triangle \triangle ABC. Points C_1 and C_2, A_1 and A_2, B_1 and B_2 are chosen at AB, BC, AC respectively such that A_1M_1 // M_2B_2 // AB, B_1M_1 // M_2C_2 // BC, C_1M_1 // M_2A_2 // AC. It is known that A_1M_1 = B_1M_1 = C_1M_1 =  \ell_1, A_2M_2 = B_2M_2 = C_2M_2 =  \ell_2. Prove that \ell_1 = \ell_2.

(Å. Tyrkevych, Chernivtsi)

P260 [also] Let A be one of the intersection points of the circles \omega_1 and \omega_2 with centres O_1,O_2. The straight line \ell is tangent to \omega_1 and \omega_2 at points B,C respectively. Denote by O_3 the circumcenter of triangle \triangle ABC. Let D be such point that A is a midpoint of O_3D. Denote by M the midpoint of O_1O_2. Prove that \angle O_1DM = \angle O_2DA.
(O. Klurman, Lviv)

P262 Let ABCD be a convex quadrangle. The incircles of triangles \triangle ABC and \triangle ABD touch AB at M and N. The incircles of triangles \triangle BCD and ACD touch CD at K and L. Prove that MN = KL.
(À. Prymak, Kyiv)

P264 Let O be the circumcenter of acute triangle \triangle AKN. The point H is chosen at side KN in arbitrary way. Let I at AN and M at AK be such that HI\perp NA and HM\perp KA. Prove that the broken line MOI bissects the area of \triangle AKN.
(I. Nagel, Evpatoria)

P266 Let ABCD be a convex quadrangle. The rays AB and DC, BC and AD intersect at points E, F respectively. The angle bisectors of \angle AED and \angle BFA intersect at K in such a way that \angle EKF = 90^o . Prove that S_{\triangle AKB} + S_{\triangle CKD} = S_{\triangle BKC} + S_{\triangle AKD}.

(O. Makarchuk, Dobrovelychkivka)

P270 Given are the circle \omega and the circle \omega_1 which touches \omega in inner way at point A. Construct the point X \ne A at \omega such that the angle between the tangent lines from X to \omega_1 is equal to the given angle.
(À. Prymak, Kyiv)

P273 Let I be the incenter of triangle \triangle ABC: The circumcircles of \triangle AIC and \triangle AIB intersect sides AB and AC respectively at points K and N. Let M be an arbitrary point of the segment KN. Prove that the sum of the distances from M to the sides of triangle \triangle ABC does not depend on the choice of point M.
(I. Nagel, Evpatoria)

P274 Let I be the incenter of triangle \triangle ABC. The straight lines AI,BI and CI intersect the outcircle \omega of triangle \triangle ABC at points D, E and F respectively. Let DK be the diameter of \omega and N be the intersection point of KI and EF. Prove that KN = IN.
(T. Tymoshkevych, Kyiv)

P276 Let ABCD be a trapezium. The circle \omega_1 with center O_1 is inscribed into the triangle \triangle ABD and the circle  \omega_2 with center O_2 touches the side CD and the extensions of the sides BC and BD of the triangle \triangle BCD. It is known that AD// O_1O_2 // BC. Prove that AC = O_1O_2.
(V. Yasinskyy, Vinnytsya)

P282 Let ABCDE be a convex pentagon such that \angle ABC = \angle CDE = 90^o and \angle BAC = \angle CED = \alpha. Let M be the midpoint of AE. Find \angle BMD.

(O. Rybak, Kyiv)

P283 [also]Construct the triangle ABC if known are the vertice A, the incenter I and the intersection point of the medians M.
(O. Makarchuk, Dobrovelychkivka)

P286 Let ABCDEF be a hexagon such that AB // CD // EF and BC// DE// FA. Prove that the straight lines AD,BE and CF are concurrent.
(E. Turkevich, Chernivtsi)

P287 Triangle ABC and point P inside it are given. Construct points A_1,B_1,C_1 at straight lines BC,AC,AB respectively such that the straight line AP bisects the segment B_1C_1, the straight line BP bisects the segment A_1C_1 and the straight line CP bisects the segment A_1B_1.

(A. Prymak, Kyiv)

P290 Diagonals AC and BD of equilateral trapezium ABCD (BC // AD, BC < AD) are orthogonal and intersect each other at point O. Let BM and CN be altitudes of trapezium. Denote by P and Q be the midpoints of OM and ON respectively. Prove that S_{\triangle ABP} + S_{\triangle 4DCQ} < S_{\triangle AOD}.

(I. Nagel, Evpatoria)

P293 Points P and Q are chosen inside the acute angle BAC in such way that PQ \perp AC. Construct with ruler and compass the point R at the side AB such that the bisector RL of triangle PQR is perpendicular to AC.
(V. Yasinskyy, Vinnytsya)


P296 The intersection line of two planes which touch the circumsphere of a tetrahedron ABCD
at points A and B is complanar with the straight line CD. Prove that \frac{AC}{BC}= \frac{AD}{BD}.

(M. Kurylo, Lypova Dolyna)

P298 Let ABCD be convex quadrangle, M be the intersection point of medians of triangle ABC and N be the point at segment MD such that MN : ND = 1 : 3 . The points E and F are chosen at straight lines AN and CN respectively in in such a way that ME //AD and MF // CD.  Prove that the straight lines AF, CE and BD are concurrent.

(T. Lazorenko, Kyiv)

P304 Let I be the incenter of triangle AB. Point D on the side AB is such that BD = BC and DC = DA. Let DM \perp AI, M \in AI. Prove that AM = MI + IC.
(I. Nagel, Evpatoria)

P306  Let ABCD be a parallelogram, P be the projection of A to BD, Q be the projection of B to AC, M and N be the orthocenters of triangles PCD and QCD respectively. Prove that PQNM is a parallelogram.
(À. Prymak, Kyiv)

P308 Point M is chosen inside the triangle ABC. The straight lines AM,BM and CM intersect sides of the triangle at points A_1,B_1 and C_1 respectively. Let K be the projection of B_1 to A_1C_1. Prove that KB_1 is a bisector of angle AKC.
(I. Nagel, Evpatoria)

P311 [also]Squares BCC_1B_2, CAA_1C_2, ABB_1A_2 are constructed from the outside at sides of triangle ABC and O_A,O_B,O_C are the centres of these squares. Let A_0,B_0,C_0 be the intersection points of the straight lines A_1B_2 and C_1A_2, A_1B_2 and B1C2, C_1A_2 and C_1A_2 respectively. Prove that the straight lines O_AA_0, O_BB_0 and O_CC_0 are concurrent.

(V. Yasinskyy, Vinnytsya)

P312 The incircle of triangle ABC with center I touches the sides AB and BC at points K and P respectively. The bissector of angle C intersects the segment KP at point Q and the straight line AQ intersects the side BC at point N. Prove that points A, I,N and B lie at a common circle.
(I. Nagel, Evpatoria)

P313 The straight line \ell intersects the side BC of triangle ABC at point X and the straight lines AC, AB at points M, K respectively. Point N is chosen at the straight line \ell in such way that AN touches the circumcircle of triangle ABC. Let L be the intersection point of the circumcircles of triangles ABC and ANX,  L \ne A. Prove that points A,M,L,K lie at a common circle.
(A. Prymak, O. Manzjuk, Kyiv)

P317 Let AB be a diameter of circle \omega. Points M, C and K are chosen at circle \omega in such a way that the tangent line to the circle \omega at point M and the secant line CK intersect at point Q and points A,B,Q are collinear. Let D be the projection of point M to AB. Prove that DM is the angle bisector of angle CDK.
(I. Nagel, Evpatoria)

P319 Circles \omega_1 and \omega_2 intersect at points A and B. Diameter BP of \omega_2 intersects the circle \omega_1 at point C and diameter BK of the circle \omega_1 intersects the circle \omega_2 at point D. The straight line CD intersects the circle \omega_1 at point S \ne C and the circle \omega_2 at point T \ne D. Prove that BS = BT.

(I. Fedak, Ivano-Frankivsk)

P321 Let \omega_1 be the circumcircle of triangle A_1A_2A_3, let W_1,W_2,W_3 be the midpoints of arcs A_2A_3,A_1A_3, A_1A_2 and let the incircle \omega_2 of triangle A_1A_2A_3 touches the sides A_2A_3,A_1A_3, A_1A_2 at points K_1,K_2,K_3 respectively. Prove that W_1K_1 +W_2K_2 +W_3K_3 \ge  2R -r where R, r are the radii of \omega_1 and \omega_2.
(A. Prymak, Kyiv)

P323 Let AA_1 and CC_1 be angle bisectors of triangle ABC (A_1 \in  BC, C_1 \in AB). Straight line A_1C_1 intersects ray AC at point D. Prove that angle ABD is obtuse.

(I. Nagel, Evpatoria)

P324 Let H be the orthocenter of acute-angled triangle ABC. Circle \omega with diameter AH and circumcircle of triangle BHC intersect at point P \ne H. Prove that the straight line AP pass through the midpoint of BC.
(Yu. Biletskyy, Kyiv)

P326 Let P be arbitrary point inside the triangle ABC, \omega_A, \omega_B and \omega_C be the circumcircles of triangles BPC, APC and APB respectively. Denote by X, Y,Z the intersection points of straight lines AP,BP, CP with circles \omega_A, \omega_B, \omega_C respectively (X, Y,Z \ne P). Prove that \frac{AP}{AX}+\frac{ BP}{BY}+ \frac{CP}{CZ}= 1.

(O. Manzjuk, Kyiv)

P329 Construct triangle ABC given points O_A and O_B,which are symmetric to its circumcenter O with respect to BC and AC, and the straight line h_A, which contains its altitude to BC.
(G. Filippovskyy, Kyiv)

P330 Let O be the midpoint of the side AB of triangle ABC. Points M and K are chosen at sides AC and BC respectively such that \angle MOK = 90^o. Find angle ACB, if AM^2 + BK^2 = CM^2 + CK^2.

(I. Fedak, Ivano-Frankivsk)

P333 Let circle \omega touches the sides of angle \angle A at points B and C, B' and C' are the midpoints of AB and AC respectively. Points M and Q are chosen at the straight line B' C' and point K is chosen at bigger ark BC of the circle \omega. Line segments KM and KQ intersect \omega at points L and P. Find \angle MAQ, if the intersection point of line segments MP and LQ belongs to circle \omega.

(I. Nagel, Evpatoria)

P335 A point O is chosen at the side AC of triangle ABC so that the circle \omega with center O touches the side AB at point K and BK = BC. Prove that the altitude that is perpendicular to AC bisects the tangent line from the point C to \omega.

(I. Nagel, Evpatoria)

P338 A circle \omega_1 touches sides of angle A at points B and C. A straight line AD intersects \omega_1 at points D and Q, AD < AQ. The circle \omega_2 with center A and radius AB intersects AQ at a point I and intersects some line passing through the point D at points M and P. Prove that I is the incenter of triangle MPQ.
 (I. Nagel, Evpatoria)

P339 The insphere of triangular pyramid SABC is tangent to the faces SAB, SBC and SAC at points G, I and O respectively. Let G be the intersection point of medians in the triangle SAB, I be the incenter of triangle SBC and O be the circumcenter of triangle SAC. Prove
that the straight lines AI, BO and CG are concurrent.
 (V. Yasinskyy, Vinnytsya)

P343 Points C_1, A_1 and B_1 are chosen at sides AB, BC and AC of triangle ABC in such a way that the straight lines AA_1, BB_1 and CC_1 are concurrent. Points C_2, A_2 and B_2 are chosen at sides A_1B_1, B_1C_1 and A_1C_1 of triangle A_1B_1C_1 in such a way that the straight lines A_1A_2, B_1B_2 and C_1C_2 are concurrent. Prove that the straight lines AA_2, BB_2 and CC_2 are concurrent.
(I. Nagel, Evpatoria)

P345 Let I be the incenter of a triangle ABC. Points P and R, T and K, F and Q are chosen on sides AB, BC, and AC respectively such that TQ//AB, RF//BC, PK//AC and the lines TQ, RF, and PK are concurrent at the point I. Prove that TK + QF + PR \ge KF + PQ + RT.

(M. Rozhkova, Kyiv)

P347 Squares ABCD and AXYZ are located inside the circle \omega in such a way that quadrilateral CDXY is inscribed into the circle \omega. Prove that AB = AX or AC \perp XY .

(O. Karlyuchenko, Kyiv)

P348 Let G be the centroid of triangle ABC. Denote by r, r_1, r_2 and r_3 the inradii of triangles ABC, GBC, GAC and GAB respectively and by p the semiperimeter of triangle ABC. Prove that \frac{1}{r_1}+ \frac{1}{r_2}+ \frac{1}{r_3}\ge\frac{3}{r} + \frac{18}{p}.

(V. Yasinskyy, Vinnytsya)

P352 Let AK, BN be the altitudes of acute triangle ABC. Points L, P are chosen at sides AB, BC such that NL \perp AB, NP \perp BC and points Q, M are chosen at sides AB, AC such that KQ \perp AB, KM \perp AC. Prove that \angle PQK = \angle NLM.

(I. Nagel, Evpatoria)

P354 Point M is chosen at the diagonal BD of parallelogram ABCD. The straight line AM
intersects the side CD and the straight line BC at points K and N respectively. Let \omega_1 be the circle with centre M and radius MA and \omega_2 be the circumcircle of triangle KNC. Denote by P and Q the intersection points of circles \omega_1 and \omega_2. Prove that the circle \omega_2 is inscribed into the angle QMP.

(I. Nagel, Evpatoria)

P358 Àn isosceles triangle has perimeter 30 sm and its orthocenter lies on the incircle. Construct
such triangle on a square 9 \times 9 sm sheet of paper. Is it possible to construct such triangle
on a smaller square sheet of paper?
(A. Lebiga, Volodarsk-Volynskyy)

P360 Let AA_1, BB_1, CC_1 be the altitudes of an acute triangle ABC. Denote by A_2, B_2 and C_2 the orthocenters in triangles AB_1C_1, A_1BC_1 and A_1B_1C respectively. Prove that the straight lines A_1A_2, B_1B_2 and C_1C_2 are concurrent.
(M. Rozhkova, Kyiv)

P362 Let \omega_1 be the incircle of a triangle ABC. The circle \omega_1 has center I and touches the sides AB and AC at points M and N. A circle \omega_2 passes through points A and I and intersects the sides AB and AC at points Q and P respectively. Prove that the line segment MN passes through the midpoint of line segment PQ.

(I. Nagel, Evpatoria)

P365 Points K and N are chosen on the side AC of a triangle ABC so that AK + BC = CN + AB. A point M is the midpoint of the segment KN and BM is the bisector of the angle ABC.  Prove that ABC is an isosceles triangle.
(I. Nagel, Evpatoria)

P367 Let ABC be an acute triangle such that \angle B = 60^o. Denote by S the intersection point of the bisector BL and altitude CD. Prove that SO = SH, where H is the orthocenter and O is the circumcenter of the triangle ABC.
(I. Nagel, Evpatoria)

P369 Let ABC be an isosceles acute triangle (AB = AC) with \angle A \ne 45^o and \omega be its circumcircle with center O. A circle \omega_1 with its center on BC passes through the points B and O and intersects the circle \omega at a point F \ne B. Prove that CF and AO intersect on \omega_1 and CF // BO.
(M. Rozhkova, Kyiv)

P370 Let ABCD be a convex quadrangle such that AB = 3, BC = 4, CD = 12, DA = 13 and S_{ACD} = 5S_{ABC}. Find S_{ABCD}.
(I. Fedak, Ivano-Frankivsk)

P373 Points A_1, B_1 and C_1 are chosen at sides AB, BC and CA of triangle ABC respectively such that AA_1 : A_1B = BB_1 : B_1C = CC_1 : C_1A = 1 : 2. Prove that P_{A_1B_1C_1} > \frac{1}{2}P_{ABC}.

(L. Orydoroga, Donetsk)

P375 The incircle of quadrangle ABCD touches the sides AB, BC, CD, DA at points K, M, N, P respectively. Points R, S are chosen at the straight line KN such that PR \perp KN, MS \perp KN. Let Q be the intersection point of the straight lines AR and BS, while T be the intersection point of the straight lines CS and DR. Prove that it is possible to inscribe a circle into the quadrangle SQRT.
(I. Nagel, Evpatoria)

P377  Medians AD and BE of a triangle ABC intersect at a point M. It is known that the
quadrilateral DCEM is both inscriptable and cyclic. Prove that ABC is an equilateral
triangle.
(I. Nagel, Evpatoria)

P379 A circle \omega intersects the side AK of a triangle AKN at points P,L (KP < KL), intersects the side KN at points H,M (KH < KM) and touches the side AN at its midpoint Q. The straight lines PH and AN intersect at a point I. Find the point K with compass and ruler, provided that only points H, I,N,A are known.
(I. Nagel, Evpatoria)

P380. A point X is chosen inside a tetrahedron ABCD. Prove that AX \cdot S_{\vartriangle BCD} + BX \cdot S_{\vartriangle ACD} + CX \cdot S_{\vartriangle ABD} + DX \cdot S_{\vartriangle ABC} \ge  9V_{ABCD}.

(S. Slobodyanyuk, Kyiv)

P383 A trapezoid ABCD is given (BC // AD). Construct with compass and ruler such points X and Y on the sides AB and CD respectively that XY // AD and YX is the angle bisector of \angle AYB.
(V. Tkachenko, Kyiv)

P385. Equilateral triangle BDM is constructed on a diagonal BD of an isosceles trapezoid ABCD (BC // AD, BC < AD, \angle A = 60^o). The side BM intersects AC and AD at points P and K respectively, CM intersects BD at a point N, O is the intersection point of diagonals AC and BD. Prove that the straight lines MO, DP and NK are concurrent.

(I. Nagel, Evpatoria)

P386. An acute angle \angle AOB and a point P inside it are given. Construct two perpendicular
segments PM and PN, where M and N lie in the rays OA and OB correspondingly, so that the rays cut from \angle AOB a quadrilateral with the maximal possible area.

(N. Beluhov, Stara Zagora, Bulgaria)

P390 Let O, H be the circumcenter and the orthocenter of triangle ABC respectively, D be the
midpoint of BC and E be the intersection point of AD and circumcircle of triangle ABC.
Construct triangle ABC if known are points D,E and the straight line OH.

(G. Filippovskyy, Kyiv)

P391 Let ABC be a triangle such that \angle A = 2\angle B \le 90^o. Find two ways of dissecting the triangle ABC into three isosceles triangles by straight cuts.
(M. Rozhkova, Kyiv)

P396 Let O be the intersection point of diagonals of rectangle ABCD. The square BKLO is
constructed on BO such that segments OL and BC intersect. Let E be the intersection
point of OL and BC. Prove that the straight lines AB, CL and KE are concurrent.

(M. Rozhkova, Kyiv)

P398 Let I_A be the center of an excircle of the triangle ABC, tangent to BC and tangent to the extensions of AC and BC. Let P and Q be the circumcenters of triangles ABI_A and ACI_A, respectively. Prove that points B, C, P and Q are concyclic.

(V. Yasinskyy, Vinnytsya)

P400 Incircle \omega of triangle ABC touches the sides AB and AC at points K and N respectively. It is known that the centroid M of this triangle lies at the segment KN. Prove that the line passing though the centroid of the triangle parallel to BC is a tangent to the circle  \omega.

(I. Kushnir, Kyiv)

P403 Let ABCD be inscribed quadrilateral. Points X and Y are chosen at diagonals AC and BD respectively such that ABXY is a parallelogram. Prove that the radii of circumcircles of
triangles BXD and AYC are equal.
(V. Yasinskyy, Vinnytsya)

P407 Let Q be the midpoint of diagonal BD of trapezium ABCD (AD // BC). It is given that AB^2 = AD \cdot BC and AQ = AC. Find BC : AD.
(M. Rozhkova, Kyiv)

P409. Let H be the intersection point of the altitudes AF and BE of acute triangle ABC, M be the midpoint of AB and MP, MQ be the diameters of circumcircles of triangles AME and BMF respectively. Prove that points P, H and Q are collinear.

(V. Yasinskyy, Vinnytsya)

P411 Let ABCD be a square. Points P and Q are chosen at sides BC and CD respectively such that \angle PAQ = 45^o. Angles \angle QAD, \angle PQC and \angle APB are in geometric progression. Find \angle QAD.
(M. Rozhkova, Kyiv)

P412 (also) Let BM be a median of isosceles triangle ABC (AC = BC). Point N is chosen at BM such that \angle BAN = \angle CBM. Prove that the angle bisector of \angle CNM is orthogonal to AN.
(V. Yasinskyy, Vinnytsya)

P414 (also) Let H and O be the orthocenter and the circumcenter of acute triangle ABC respectively. It is known that AB < BC. Straight line BO intersects AC at point P, while straight line through H parallel to BO intersects AC at point Q. Prove that OP = OQ.

(V. Yasinskyy, Vinnytsya)

P418 Points M and N are chosen inside triangle ABC such that point M lies inside triangle ABN and point N lies inside triangle ACM. Moreover \angle MAB  =\angle NAC, \angle MBA = \angle NBC and \angle MCB = \angle NCA. Prove that if points B,M,N and C belong to a circle with center W then the straight line AW bisects MN.

(V. Yasinskyy, Vinnytsya)

P419. Let H and O be orthocenter and circumcenter of triangle ABC. It is known that \angle BAO = \frac{1}{3} \angle BAC and CO = CH. Determine the angles of triangle ABC.

(M. Rozhkova, Kyiv)

P422 A semicircle is given with diameter AB. On arc AB of the semicircle, an arbitrary point C is chosen that differs from points A and B. Let D be orthogonal projection of point C on the diameter AB. A circle \omega touches segments AD, CD and arc AB at point P. Prove that the intersection point of bisectors of angles \angle APB and \angle ACD lies on the diameter AB.

(V. Yasinskyy, Vinnytsya)

P426 Triangle ABC is given. Point M moves along the side BA, and point N moves along the extension of the side AC after point C in such a way that BM = CN. Prove that the
circumcenter of triangle AMN moves along a straight line.
(V. Yasinskyy, Vinnytsya)

P427  (also)Let CC_1 be the angle bisector and I the incenter of triangle ABC, \angle A = 60^o, \angle B = 80^o. Prove that P_{\vartriangle BIC_1} = BC.
(M. Rozhkova, Kyiv)

P429 Acute triangle ABC is given. Let \omega be a circle that intersects the side AB in points C_1 and C_2 (AC_1 < AC_2), the side BC in points A_1 and A_2 (BA_1 < BA_2), and the side CA in points B_1 and B_2 (CB_1 <  CB_2). Prove that \omega can be chosen in such a way that segments A_1B_2, B_1C_2 and C_1A_2 are its diameters.

(V. Yasinskyy, Vinnytsya)

P430 Let ABCD be a quadrilateral with three equal sides AB = BC = CD, O is the intersection point of the diagonals, OE \perp BC, M and N are midpoints of diagonals AC and BD respectively. Prove that O is the incenter of triangle EMN.
(M. Rozhkova, Kyiv)

P432 A circumscribed trapezium ABCD (BC // AD) is such that CD = \frac{2BC \cdot AD}{BC+AD}. Find angle \angle ADC.
(I. Kushnir, Kyiv)

P436 Let I be incenter and r be inradius of triangle ABC. Circle \omega with center I and radius 2r intersects sides AB and AC at points D and E respectively. Moreover DE is a diameter of \omega. Find \angle BAC.
(M. Rozhkova, Kyiv)

P439 (also)A triangle ABC is given with AB > AC. A tangent to the circumcircle of triangle ABC at point A intersects the line BC at point P. Point Q is chosen at the extension of BA beyond A such that AQ = AC. Let X and Y be the midpoints of segments CQ and AP respectively, R is chosen at the segment AP such that AR = CP. Prove that CR = 2XY.

(V. Yasinskyy, Vinnytsya)

P441 A point P is located in the plane of convex quadrangle ABCD, Let A_0, B_0, C_0 and D_0 be midpoints of AB,BC,CD and DA respectively. A point A_1 is chosen at side AB such that rays PA_0 and PA_1 are symmetric with respect to the angle bisector of \angle APB. Points B_1,C_1 and D_1 are chosen in a similar way at sides BC,CD and DA respectively. Find all points P for which quadrangle A_1B_1C_1D_1 is a parallelogram.

(V. Yasinskyy, Vinnytsya)

P444 Let AA_1, BB_1, CC_1 be the altitudes of acute triangle ABC. Let AK, BL and CM be the perpendiculars drawn from points A,B and C to the straight lines A_1B_1, B_1C_1 and C_1A_1 respectively. Prove that A_1K = B_1L = C_1M.
(M. Rozhkova, Kyiv)

P446 Let M be arbitrary point inside triangle ABC and N be arbitrary point of the segment AM. Straight lines AB and AC intersect the circumcircle of triangle BMC for the second time at points E and F respectively. Straight line EM intersects the circumcircle of triangle NMC for the second time at point P, while straight line FM intersects the circumcircle of triangle NMB for the second time at point Q. Prove that the circumcircles of triangles EMF and PMQ touch each other.
(V. Yasinskyy, Vinnytsya)

P448 Let AL be an angle bisector of triangle ABC such that AB > AC. Point K on the side AB is such that AK = AC. Let N be the intersection point of circumcircles of triangles ABC
and CKL. Prove that points A, L,N are collinear.
(O. Tarasyuk, Kyiv)

P450 Let E be an arbitrary point on the side AC of triangle ABC. Points N and M are chosen on the rays AB and CB respectively such that \angle AEN = \angle ABC = \angle CEM. Rays AM and CN intersect at point K. Prove that when point E varies, line KE passes through a fixed point.

(V. Yasinskyy, Vinnytsya and I. Nagel, Evpatoria)

P452 In the triangle ABC let AD be the diameter of a circumcircle, H be the orthocenter and
E be the midpoint of AH. Construct triangle ABC if points D,E and line \ell which contains BC are given.
(S. Yakovlev, Kyiv)

P455 Points D and E lie in the interior of an angle A. Construct points B and C on sides of the angle such that D lies on the segment BC and E lies on a circumcircle of triangle ABC.

(Ye. Diomidov, Kyiv)

P459 Let ABCD be a quadrilateral inscribed in a circle of diameter BD and M be an arbitrary
point on the shorter arc AD. Let MN, MK, MP, MT be perpendiculars from M onto lines
AB, BC, CD, AD respectively. Prove that S_{\vartriangle MNP} = S_{\vartriangle MKT}

(I. Nagel, Evpatoria)

P460 Let AH be the altitude of acute triangle ABC. Construct triangle ABC if BH, CH and
AB + AC are given. 
(A. Nikolayev, Kyiv)

P462 Let the tangents to the circumcircle of a triangle ABC at vertices B and C intersect at point D and let E be the point of intersection of AD and BC. Prove that AE = ED if and only if AB^2 + AC^2 = 2BC^2
(V. Brayman, Kyiv)

P463 Let AD be the altitude of acute triangle ABC, O be the circumcenter and H be the
orthocenter of this triangle, MN be the midline parallel to BC, and T be the intersection point 
of AO and MN. Prove that the midpoint of OH belongs to TD.
(O. Karlyuchenko, Kyiv)

P466 Incircle of triangle ABC with center I touches sides BC,AC,AB at points  K_1,K_2,K_3. Straight lines AI and CI intersect the segment K_1K_3 at points E and F. Straight lines AF and CE intersect at point T. Prove that points K_2, I, T are collinear. 

(M. Rozhkova, Kyiv)

P470 Straight line parallel to side BC of triangle ABC intersects sides AB and AC at points
P and Q respectively. Point M is chosen arbitrarily inside triangle APQ. Segments BM
and CM intersect the segment PQ at points E and F respectively. Let N be the second
intersection point of circumcircles of triangles PMF and QME. Prove that points A, M
and N are collinear. 
(V. Yasinskyy, Vinnytsya)

P473 Let BH_2 and CH_3 be the altitudes of triangle ABC. Restore the triangle if the point A and the lines BC, H_2H_3 are given. 
(S. Yakovlev and G. Filippovskiy, Kyiv)

P475 In a non-equilateral triangle ABC it is given that AB^2 +BC^2 = 2AC^2. Let AT and CP be the altitudes, H be the orthocenter, and M be the intersection point of the medians of the triangle ABC. Prove that the lines AC, PT and HM are concurrent. 
(A. Trygub, Kyiv)

P476 Let the diagonals of a convex quadrilateral ABCD intersect at point L, and it holds AB =AC = BD. Let P be the second intersection point of circumcircles of triangles ABC and ALD, and the lines BC and AP intersect at point Q. Prove that LQ is angle bisector of the angle \angle CLD

(V. Yasinskyy, Vinnytsya)

P478 Point A lies on the circle \omega and point X lies inside or outside this circle. Construct points B and C on \omega such that point X is the center of incircle or circumcircle of triangle ABC.
(I. Kushnir, Kyiv)

P480 Quadrangle ABCD is inscribed into circle with diameter AD, Let K be the intersection
point of diagonals of ABCD. Circle \omega with center K touches BC. Tangent lines drawn to \omega from points B and C intersect at point N. Prove that N lies on AD.
(O. Karlyuchenko, Kyiv)

P482 Let \omega be the circumcircle of triangle ABC, \ell be the tangent line to the circle \omega at point A. The circles \omega_1 and \omega_2 touch lines \ell, BC and circle \omega externally. Denote by D, E the points where circles \omega_1, \omega_2 touch BC. Prove that the circumcircles of triangles ABC and ADE are tangent. 
(M. Plotnikov, Kyiv)

P484. In the pentagon ABCDE it is known that BC //AE, BC = \frac{1}{2}AE, DE //AB and DE = \frac{1}{2}AB. Prove that CD //BE and CD = \frac{1}{2} BE.
(O. Gryschenko, Kyiv)

P486. In the triangle ABC let O be the circumcenter, H be the orthocenter and E be the midpoint of OH. Construct triangle ABC if lines BC, AO and point E are given.

(K. Kadirov and K. Yatzkiv, Kyiv)

P488 Let n > 1 be positive integer. Point A_1 is chosen inside triangle ABC such that \angle ABA_1 =\frac{1}{n}\angle ABC and \angle ACA_1 = \frac{1}{n}\angle ACB. Points B_1 and C_1 are defined in similar way. Prove that the straight lines AA_1, BB_1 and CC_1 are concurrent. 
(V. Yasinskyy, Vinnytsya)

P493 Trapezium ABCD (BC //AD, BC < AD) is inscribed into the circle \omega. Let M be the midpoint of AD, straight line CM intersects \omega at point T, X be the midpoint of BT, straight line AX intersects \omega at point Y. Prove that DY // BT

(T. Batsenko, Kyiv)

P496 Point T is symmetric to the center of square ABCD with respect to the point A. Reconstruct the square if known are points B and T.
(D. Kravetz, Kyiv)

P498 Angle bisector of angle \angle A of triangle ABC intersects the circumcircle at point W. Straight line  \ell  // AC passes through point W and intersects AB and BC at points P and K respectively. It is known that AK = CP. Prove that BP = KW
(O. Baranovskiy, Kyiv)

P500 Let D be an arbitrary point on the side BC of acute triangle ABC, Perpendicular bisector of segment BD intersects AB at point X, and perpendicular bisector of segment DC intersects AC at point Y . The circumcircle of triangle DXY intersects the side BC again at point Z. Prove that the orthocenter of triangle XY Z does not depend on the choice of point D.

(D. Khilko, Kyiv)

P503 In triangle ABC the orthocenter H, the circumcenter O and excenter I_a are collinear. Is it necessarily true that triangle ABC is isosceles?
(I. Kushnir, Kyiv)

P504 Three hedgehogs were in the vertices of equilateral triangle with side length 100 m. Then the first hedgehog strolled  1 m along the straight line, the second hedgehog strolled 2 m and the third hedgehog strolled 3 m (maybe along different straight lines). Is it possible that the hedgehogs are in the vertices of
a) equilateral triangle?
b) equilateral triangle with side length 100 m?
(O. Tolesnikov, Jerusalem)

P508. Circles \omega_1 and \omega_2 intersect at points A and B. Points D, H are chosen on the circle \omega_1 and points E, G are chosen on the circle \omega_2 such that points D, A, E are collinear, DG is a tangent line to \omega_2 and EH is a tangent line to \omega_1. Prove that the segments DE, DG and EH are sides of a right triangle.

(M. Plotnikov, Kyiv)

P509. Point T is chosen on chord AB of a circle with center O. Let K be the foot of a perpendicular drawn from point T on OB and Q be the circumcenter of triangle ATK. Prove that OQ // AB.
(M. Vlasenko, Kyiv)

P516 Let BT be the altitude and H be the intersection point of the altitudes of triangle ABC. Point N is symmetric to H with respect to BC. The circumcircle of triangle ATN intersects BC at points F and K. Prove that FB = BK.
(V. Starodub, Kyiv)

P520 Let O be the center of circle \omega, let KA and KB be tangent lines to \omega and let Q be an arbitrary point on the chord AB. Straight line \ell \perp OQ passes through point Q and intersects KA, KB at points E, F respectively. Prove that Q is the midpoint of EF.

(A. Shapoval, Kyiv)

P522 Let D be an interior point of triangle ABC. Angle bisectors of angles \angle BAC and \angle ACD intersect at point N. Angle bisector of angle \angle ABD and straight line which contains angle bisector of angle \angle BDC intersect at point T. Let Q be the intersection point of straight lines AB and CD. Prove that points N,T,Q are collinear.

(O. Karlyuchenko, Kyiv)

P530 Let ABCD be an isosceles trapezium (AD // BC). Points K and N are chosen on the sides AB and CD such that AK = CN. The segment KN intersects the diagonals AC and BD at points S and T respectively. Prove that the circumcircles of triangles AKS, BKT, CNS and DNT have a common point.
(V. Brayman, Kyiv)



sources: aopsofficial page (old)

P.S. Starting form 2017, it has become online magazine with no English translation available, and without the above problem archive available.


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