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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 

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) and  (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 $C$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 $A_3B_3$ have 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 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 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 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 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 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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