In the World of Mathematics

geometry problems with aops links from 

U sviti matematyky  /   In the World of Mathematics

                                                       Ukrainian Magazine Magazine 

from this magazine with aops links in their problem number after I propose them there,

 geometry problems withs aops links
collected inside aops here


                                          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 [alsoPoints $A, B, C$ 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 $C$D 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 $A_3B_3$ ave a common point.


(O. Manzjuk, Kyiv)

P139 A circle inscribed in a triangle, $ABC$, touches the sides A$B, BC$ 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 [also]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 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  [also]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 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 [also]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 B$C$ 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 [also]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 [also]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 [also]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 [also]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  [also]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 A$BCDE$ 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 midsegment of 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 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 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)

426 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)

427 Let $CC_1$ be the angle bisector and $I$ the incenter of triangle $ABC, \angle A = 60^o, \angle B = 80^o$. Prove that $P_{\vartriangle B_IC_1} = BC$.

(M. Rozhkova, Kyiv)

429 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 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 excircle 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)

484. 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 $CD = \frac{1}{2} B$E.

(O. Gryschenko, Kyiv)

486. 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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