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U sviti matematyky - geometry part I

3

geometry problems from ''In the World of Mathematics'' (U sviti matematyky)

26

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 feet of the perpendiculars, that are drawn from the points $A$ and $C$ on the chords $CM$ and $AM$ correspondingly. Prove, that the area of $KPMQ$ equals $S$ .

(I. Nagel, Eupatoria)

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41

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)

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48

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 $\frac{MK}{KN}=\frac{TM}{TN}.\frac{BD}{AC}$ .

tdl

51

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

(I. Nagel, Herson)

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53

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 intersection point 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)

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55

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

(O. Kukush, Kyiv)

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59

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)

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61

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 fixed 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)

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68

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

(V. Yasinskyy, Vinnytsa)

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73

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)

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79

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)

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96

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)

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97

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)

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103

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

(V. Yasinskyi, Vinnytsa)

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104

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)

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109

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)

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112

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.)

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114

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)

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116

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.)

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118

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)

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122

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

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128

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)

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133

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)

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134

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)

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139

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)

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140

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)

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143

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)

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153

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, Vinnytsa)

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154

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)

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157

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.)

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168

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.)

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173

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)

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178

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.)

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181

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?

rogue

186

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.)

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187

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)

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191

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)

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193

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.)

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201

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.)

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204

In convex pentagon $ABCDE$ , $\angle ABC=\angle AED=90^{\circ}$ and $AB\cdot DE=BC\cdot AE$ . Let $F$ be the intersection of $CE$ and $BD$ .

Prove that $AF$ is perpendicular to $BE$

discredit

206

The diagonals $AC,BD$ of convex quadrilateral $ABCD$ cut at $P$ . The circumcircles of triangles $ABP,DCP$ cut at $M$ distinct from $P$ . The circumcircles of triangles $ADP,BCP$ cut at $N$ distinct from $P$ . Perpendiculars to $AC,BD$ passing through midpoints of $AC,BD$ respectively meet at $O$ .

Prove that $M,O,P,N$ are concyclic.

discredit

210

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)

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212

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.)

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218

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.)

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221

Point $P$ is chosen inside triangle $ABC$ . Let $X,Y,Z$ be the intersections of $AP,BP,CP$ with $BC,CA,AB$ respectively. Let $M_1,M_2,M_3$ be the midpoints of $AC,AB,BC$ and $N_1,N_2,N_3$ the midpoints of $XZ,XY,YZ$ respectively.

Prove that $M_1N_1,M_2N_2,M_3N_3$ concur.

discredit

230

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

Pirkuliyev Rovsen

235

The incircle of triangle $ABC$ touches $BC,CA,AB$ at $K,L,M$ respectively. The perpendiculars at $K,L,M$ to $LM,KM,KL$ interesect the incircle again at $P,Q,R$ respectively.

Prove that $AP,BQ,CR$ concur.

discredit

240

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)

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243

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)

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245

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)

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247

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?

rogue

249

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)

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252

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)

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254

The circle $\omega$ passing through the vertices $B,C$ of triangle $ABC$ with $AB\neq AC$ intersects the sides $AB,AC$ at $R,S$ respectively. Let $M$ be the midpoint of $BC$ . The line perpendicular to $MA$ at $A$ intersects $BS,CR$ at $K,T$ respectively. Suppose that $TA=AK$ .

Prove that $MS=MR$ .

discredit

257

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)

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260

Let $A$ be one of the intersection points of circles $w_1,w_2$ with centers $O_1,O_2$ . The line $l$ is tangent to $w_1,w_2$ at $B,C$ respectively. Let $O_3$ be the circumcenter of triangle $ABC$ . Let $D$ be a point such that $A$ is the midpoint of $O_3D$ . Let $M$ be the midpoint of $O_1O_2$ .

Prove that $\angle O_1DM=\angle O_2DA$ .

discredit

262

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)

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264

Let $O$ be the circumcenter of acute triangle $AKN$ . $H$ is an arbitrary point on side $KN$ . Let $I$ on $AN$ and $M$ on $AK$ be such that $HI$ is perpendicular to $NA$ and $HM$ is perpendicular to $KA$ . Prove that the broken line $MOI$ bisects the area of triangle $AKN$ .

discredit

266

In convex quadrilateral $ABCD$ , let $E=AB\cap DC, F=BC\cap AD$ . The angle bisectors of $\angle AED,\angle BFA$ intersect at $K$ . Given that $\angle EKF=90^{\circ}$ , prove that

$[AKB]+[CKD]=[BKC]+[AKD]$ , where

$[...]$ denotes the area of triangle $...$

discredit

270

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)

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273

Let $I$ be the incenter of triangle $ABC$ . The circumcircles of triangles $AIC,AIB$ intersect sides $AB,AC$ at $K,N$ respectively. Let $M$ be an arbitrary point on segment $KN$ . Prove that the sum of the distances from $M$ to the sides of triangle $ABC$ does not depend on the choice of $M$ .

discredit

274

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)

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276

Let $ABCD$ be a trapezoid. The circle $w_1$ with center $O_1$ is inscribed into the triangle $ABD$ , and the circle $w_2$ with center $O_2$ touches side $CD$ and the extensions of sides $BC$ and $BD$ of triangle $BCD$ . It is known that $AD||O_1O_2||BC$ .

Prove that $AC=O_1O_2$ .

discredit

282

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)

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283

Construct the triangle $ABC$ if known are the vertex $A$ , the incenter $I$ and the intersection point of the medians $M$ .

(O. Makarchuk, Dobrovelychkivka)

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286

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)

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287

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)

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290

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)

parmenides51

293

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)

parmenides51

296

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)

parmenides51

298

Let $ABCD$ be a convex quadrilateral. $M$ is the centroid of triangle $ABC$ , and $N$ is on segment $MD$ such that $MN: ND = 1: 3$ . The points $E,F$ are on $AN,CN$ respectively such that $ME||AD,MF||CD$ . Prove that $AF,CE,BD$ concur.

discredit

304

Let I be the incenter of triangle ABC.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 .

_________________________________________
Azerbaijan Land of Fire

Pirkuliyev Rovsen

306

Let $ABCD$ be a parallelogram, $P$ the projection of $A$ on $BD$ , $Q$ the projection of $B$ on $AC$ . Let $M,N$ be the orthocenters of triangles $PCD,QCD$ respectively. Prove that $PQNM$ is a parallelogram.

discredit

308

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)

parmenides51

311

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 $B_1C_2,$ $C_1A_2$ and $C_1A_2$ respectively. Prove that the straight lines $O_AA_0,$ $O_BB_0$ and $O_CC_0$ are concurrent.

rogue

312

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.

rogue

313

The straight line $l$ 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 $l$ 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.

rogue

317

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 $CDK.$

rogue

319

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 $BS=BT.$

rogue

321

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 $W_1K_1+W_2K_2+W_3K_3\ge2R-r,$ where $R, r$ are the radii of $\omega_1$ and $\omega_2.$

rogue

323

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.

rogue

324

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 $BC.$

rogue

326

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 $\frac{AP}{AX}+\frac{BP}{BY}+\frac{CP}{CZ}=1.$

rogue

329

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 $BC.$

rogue

330

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 $\angle MOK=90^{\circ}.$ Find angle $ACB,$ if $AM^2+BK^2=CM^2+CK^2.$

rogue

333

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 $\omega.$

rogue

335

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 $\omega$ .

rogue

338

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 $MPQ.$

rogue

339

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.

rogue

343

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.

rogue

345

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 $TK+QF+PR\ge KF+PQ+RT.$

rogue

347

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 $AC\perp XY$ .

rogue

348

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 $\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}\ge\frac{3}{r}+\frac{18}{p}.$

rogue

352

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 $\angle PQK=\angle NLM.$

rogue

354

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 $QMP.$

rogue

358

Àn isosceles triangle has perimeter $30$ sm and its orthocenter lies on the incircle. Construct such triangle on a square $9\times9$ sm sheet of paper. Is it possible to construct such triangle on a smaller square sheet of paper?

rogue

360

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.

rogue

362

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 $PQ.$

rogue

365

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)

parmenides51

367

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)

parmenides51

369

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)

parmenides51

370

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)

parmenides51

373

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)

parmenides51

375

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)

parmenides51

377

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)

parmenides51

379

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)

parmenides51

380

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)

parmenides51

383

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)

parmenides51

385

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)

parmenides51

386

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)

parmenides51

390

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)

parmenides51

391

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)

parmenides51

396

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)

parmenides51

398

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)

parmenides51

400

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)

parmenides51

403

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)

parmenides51

409

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)

parmenides51

411

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)

parmenides51

412

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)

parmenides51

407

Let $Q$ be the midpoint of diagonal $BD$ of trapezium $ABCD$ ( $AD{\parallel}BC$ ).It is given that $AB^2=AD{\cdot}BC$ and $AQ=AC$ .Find $BC:AD$ .

Pirkuliyev Rovsen

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