Kukin MO - OmSU 2008-22 VII+ (Russia) 85p

geometry problems from Kukin Math Olympiad , by Omsk State University (Russia) with aops links

2008 - 2022

collected inside aops here

2008 Kukin MO - Omsk State University 7.3

The plane is drawn into equilateral triangles as shown. Find the value of the angle $ABC$.

(Usov S.)

2008 Kukin MO - Omsk State University 8.3

Points $D, E, F$ were chosen on the sides $AB, BC, CA$ of triangle $ABC$ respectively so that $EF\parallel AB$ and $ED\parallel AC$. Lines $DF$ and $BC$ intersect at point $K$. It turned out that $DF = FK$. Find the ratio $BE: EC$.

2008 Kukin MO - Omsk State University 9.2

On the side $BC$ of triangle $ABC$, point $K$ is chosen so that the angle $SAK$ is half the angle $ABC$, and the point of intersection $O$ of the segment $AK$ with bisector $BL$ of angle $B$ divides this segment into two equal parts. Prove that $AO\cdot LC = BC\cdot OL$.

(Usov S.V.)

2008 Kukin MO - Omsk State University 9.5

The vertices of a cyclic quadrilateral are connected by segments with some point inside it, thus the quadrilateral is divided into $4$ triangles. One of these triangles is known to be equilateral, the second is isosceles (not equilateral), and the other two are right . Prove that the right triangles are congruent.

(Usov S.V.)

2008 Kukin MO - Omsk State University 10.2

In the quadrangle $ABCD$, the angle $D$ is acute, and the angle $A$ is obtuse. It is known that $CD = 2AB$ and $S_{ACD}=2S_{ABD}$ . Find the ratio $\frac{S_{AOB}}{S_{COD}}$, where $O$ is the intersection point of the diagonals of the quadrilateral.

(Usov S.V.)

2008 Kukin MO - Omsk State University 11.2

An arbitrary triangle $ABC$ is given. A sequence is built by triangles according to the following rule. Three sides of triangle $A_1B_1C_1$ are equal to the sines of the angles of triangle $ABC$, three sides of triangle $A_2B_2C_2$ are equal to the sines of the angles of the triangle $A_1B_1C_1$, etc. Find the ratio of the area of triangle $A_1B_1C_1$ to the area of triangle $A_{2007}B_{2007}C_{2007}$.

(Usov S.V.)

2008 Kukin MO - Omsk State University 11.6

Octahedron $ABCDKLMN$ is inscribed in a sphere, its edges are $ABCD$ and $KLMN$ lie in parallel planes, and $ABLK$, $BCML$, $CDNM$, $DAKN$ are its the rest of the faces. It is known that $AB \cdot LM = BC\cdot KL$. Prove that octahedron $ABCDKLMN$ is either a truncated pyramid or a right prism.

(Usov S.V.)

2009 Kukin MO - Omsk State University 7.4

$OA$ and $OB$ beams form a right angle. Inquisitive seventh grader Petya held rays $OC$ and $OD$ inside this angle, forming an angle of $10^o$, and then counted all acute angles between any pairs of drawn rays (not only neighboring). It turned out that the sum of the largest and smallest of the angles found is $85^o$. Find the values of the three angles by which the right angle is split by the beams $OS$ and $OD$.

(Stern A.)

2009 Kukin MO - Omsk State University 8.1

The length of the rectangle has been decreased by $10 \%$ and the width has been decreased by $20 \%$. In this case, the perimeter of the rectangle has decreased by $12 \%$. By what percentage will the perimeter of the rectangle decrease if its length is reduced by $20 \%$, and reduce the width by $10 \%$?

2009 Kukin MO - Omsk State University 8.3

Can the distances from a point on the plane to the vertices of a certain square be equal to $1, 1, 2$ and $3$?

2009 Kukin MO - Omsk State University 9.3

In an cyclic quadrilateral, each side has length either $6$ or $8$, and one of the diagonals is length $10$. Find the radius of it's circumscribed circle.

(Usov S.V.)

2009 Kukin MO - Omsk State University 9.5

In a convex quadrilateral $ABCD$, $AB = 10$, $BC = 12$, $BD = 15$, $\angle A = \angle D$ and $\angle ABD = \angle BCD$. Find the length of the segment $CD$

2009 Kukin MO - Omsk State University 10.3

$AA1, BB1, CC_1$ are the medians of triangle $ABC$. $O_1O_2O_3$ are the points the intersections of the medians of triangles $AA_1B$, $BB_1C$, and $CC_1A$, respectively. Find the area of triangle $O_1O_2O_3$ if the area of triangle $ABC$ is equal to $1$.

(Kukina E.G.)

2009 Kukin MO - Omsk State University 11.3

Given a cube $ABCDA_1B_1C_1D_1$ with edge $1$. Line $\ell$ passes through point $E$, midpoint of edge $C_1D_1$, and intersects lines $AD_1$ and $A_1B$. Find the distance from point $E$ to the point of intersection of line $\ell$ with line $A_1B$.

2010 Kukin MO - Omsk State University 7.6

There are three sticks that make up a triangle. It is permitted to make a new triangle by breaking off identical pieces from any two sticks and gluing them to the third. Seventh grader Petya is sure that by acting in this way many times, it is possible to achieve that the triangle becomes equilateral. Is he right?

(Usov S.)

2010 Kukin MO - Omsk State University 8.2

There are four properties of quads:

(1) opposite sides are equal in pairs;

(2) two opposite sides are parallel;

(3) some two adjacent sides are equal;

(4) the diagonals are perpendicular and divided by the point of intersection in the same ratio.

One of these two quadrilaterals has some two of these properties, the other two others.

Prove that one of these two quads is a rhombus.

2010 Kukin MO - Omsk State University 9.3

Right-angled triangle $ABC$ (leg $CB$ is larger than leg $AC$), inscribed in circle. On side $BC$, point $D$ is chosen such that $AC = BD$, point $E$ is midpoint of arc $ACB$. Find the angle $CED$.

2010 Kukin MO - Omsk State University 10.4

A semicircle is constructed with diameter $AC$, the larger leg of a right-angled triangle $ABC$ with a right angle $C$, that intersects the hypotenuse $AB$. On the semicircle a point $P$ is selected such that $CP = BC$, and on the leg $AC$ a point Q is selected such that $AQ = AP$. The segment $BQ$ meets the semicircle at point $S$. Prove that $\angle  CSP = 2\angle CBP$.

2010 Kukin MO - Omsk State University 11.5

Diagonals drawn from any vertex of the $n$-gon divide the angle at this vertice into equal parts. For what values of $n$ exists such an $n$-gon that is not regular?

2011 Kukin MO - Omsk State University 7.5

Petya drew $3$ red and $3$ blue lines, and marked those points intersections through which straight lines of different colors pass. Could he find that exactly half of all intersection points are marked?

(Shapovalov A.V.)

2011 Kukin MO - Omsk State University 8.2

Divide a $30 ^ \circ$ right-angled triangle into two smaller triangles so that some median of one of these triangles is parallel to one of the angle bisectors of the second triangle.

2011 Kukin MO - Omsk State University 8.6

Given a pentagon $ABCDE$ such that $AB = BC = CD = DE$, $\angle B = 96 ^ \circ$ $\angle C = \angle D = 108 ^ \circ$. Find $\angle E$.

2011 Kukin MO - Omsk State University 9.2

The altiude $AH$ and the angle bisector $CK$ of the triangle $ABC$ divide it into four parts, two of which isosceles triangles. Find the ratio $AC: BC$ .

(folklore)

2011 Kukin MO - Omsk State University 9.4

In an isosceles triangle, the incircle touches the base $AB$ at point $F$, and the lateral sides $BC$ and $AC$ , at points $D$ and $E$, respectively. Line segment $AD$ intersects the circle at point $Q$. Prove that line $EQ$ bisects $AF$.

(folklore)

2011 Kukin MO - Omsk State University 10.1

Points $A, B, C, D$ are adjacent vertices of a regular polygon (in that order). It is known that $ACD = 120^o$. How many vertices does this polygon have?

2011 Kukin MO - Omsk State University 10.5

Point $P$ is taken inside the triangle $ABC$ so that $\angle BPC=90^o$ and $\angle BAP=\angle BCP$. Points $M$ and $N$ are the midpoints of sides $AC$ and $BC$, respectively, with $BP=2PM$. Prove that points $A, P$ and $N$ lie on the same line.

(folklore)

2011 Kukin MO - Omsk State University 11.3

Is there a convex polyhedron other than a pyramid in which the sum of the plane angles at two vertices is equal to the sum of the flat angles at the other vertices?

(Stern A.S.)

2011 Kukin MO - Omsk State University 11.5

The diagonals of the cyclic quadrilateral $ABCD$ intersect at point $X$. Points $K, L, M, N$ are feet of perpendiculars drawn from point $X$ on sides $AB, BC, CD$ and $DA$, respectively. Prove that $KL + MN = LM + KN$.

(folklore)

2012 Kukin MO - Omsk State University 7.4

A quadrilateral $ABCD$ is drawn on checkered paper. What is the sum of the measures of the angles $A$ and $C$ in degree ?

(Usov S.V.)

2012 Kukin MO - Omsk State University 8.5

A hexagon was drawn on checkered paper and partially painted over with gray (see figure). Which part of the hexagon has a large area: filled or unfilled? Justify your answer.

(Chernyavskaya I.A.)

2012 Kukin MO - Omsk State University 9.4

In a cyclic pentagon, one of the diagonals of each corner is the bisector of the angle between the side and the other diagonal. Prove that there are four equal angles in the pentagon bounded by the diagonals.

(Usov S.V.)

2012 Kukin MO - Omsk State University 10.3

Line $\ell$ touches some circle at point $A$. $P$ is a point diametrically opposite to point$A$. Another circle is drawn, which externally touches the first circle, touches line $\ell$, and lies on the same side of line $\ell$ as the first circle . Prove that the length of the tangent segment drawn from point $P$ to the second circle does not depend on the radius of the second circle.

(folklore)

2012 Kukin MO - Omsk State University 11.2

The lengths of the sides of the triangle form an arithmetic progression with a nonzero difference. Prove that exactly one angle of this triangle is greater than $60$ degrees.

(Stern A.S.)

2012 Kukin MO - Omsk State University 11.5

Given a cube $ABCDA_1B_1C_1D_1$. Points $N_1, M_1, P_1$ are selected on edges $A_1B_1, A_1D_1$ and $AA_1$, and points $N_2, M_2, P_2$ are selected on edges $CD, CB$ and $CC_1$, respectively. The distance between straight lines $N_1M_1$ and $N_2M_2$ is equal to $a$, the distance between straight lines $M_1P_1$ and $M_2P_2$ is equal to $b$, the distance between straight lines $N_2P_2$ and $N_1P_1$ is equal to $c$. It is known that the numbers $a, b, c$ are pairwise distinct. Prove that lines $P_1P_2, M_1M_2$ and $N_1N_2$ meet at one point.

(folklore)

2013 Kukin MO - Omsk State University 7.4

The square kingdom of Dania with a side of $100$ km consists of two rectangular (non-square) counties Anya and Bania, and two square counties, Vania and Gania. Neither side of these counties exceeds $90$ km. Prove that the rectangles Anya and Bania are equal.

Oral given note: Square counties, Vania and Gania, have different sizes.

2013 Kukin MO - Omsk State University 8.2

A smaller square was cut out of the square, one of the sides of which lies on the side of the original square. The perimeter of the resulting octagon is $40 \%$ larger than the perimeter of the original square. How many percent is its area less than the original square?

2013 Kukin MO - Omsk State University 8.4

In parallelogram $ABCD$ with side $AB = 1$, point $M$ is the midpoint of side $BC$, and the angle $AMD$ is $90$ degrees. Find the side $BC$.

2013 Kukin MO - Omsk State University 9.4

Triangle $KMN$ is cut from triangle $ABC$ so that segment $KM$ lies on one side $AB$, and lines $KN$ and $MN$ are parallel to sides $BC$ and $AC$, respectively. Perimeter of the resulting hexagon is $20\%$ larger than the perimeter of triangle $ABC$. If we cut the same triangle, attaching side $KN$ to side $BC$, then the perimeter of the hexagon will be $30\%$ larger than the perimeter of triangle $ABC$. And if you cut out the same triangle, attaching side $MN$ to side $AC$, then the perimeter of the hexagon will be $40\%$ larger than the perimeter of the triangle $ABC$. Which percentage of the area of the resulting hexagon is less than the area of the original triangle?

2013 Kukin MO - Omsk State University 9.5

Point $P$ is chosen inside triangle $ABC$ so that $\angle PAC = \angle PBC$. From point $P$ are drawn perpendiculars $PM$ and $PL$ on sides $AC$ and $BC$, respectively. Prove that the midpoint of side $AB$ is equidistant from points $M$ and $L$.

2013 Kukin MO - Omsk State University 10.3

In a right-angled triangle, a median was drawn from one acute angle, and from the other an angle bisector. The median divided the angle bisector in a ratio of $3: 2$, counting from the vertice. Find the acute angles of the triangle.

2013 Kukin MO - Omsk State University 10.5

The extensions of the sides $AB$ and $CD$ of the cyclic quadrilateral $ABCD$ intersect at the point $X$.The extensions of the sides $AD$ and $BC$ intersect at the point $Y$. The bisector of angle $AYB$ intersects side $AB$ at point $G$, the bisector of angle $BXC$ meets $AD$ at point $E$. Prove that line $EG$ parallel to the diagonal $BD$.

2013 Kukin MO - Omsk State University 11.2

One of the altitudes of the triangle is equal to the arithmetic mean of the other heights. Prove that the ratio of any two sides of this triangle is greater than $2/5$.

2013 Kukin MO - Omsk State University 11.5

Santa Claus and Snow Maiden are packing firecrackers in holiday boxes. Box is the right quadrangular pyramid, whose all edges are equal to $1$. Santa Claus puts the cracker in the box, and Snow Maiden lays down cracker to the bottom of the box by diagonals as shown in figure. Moreover, in both cases, the cracker is packed tightly: it does not move inside boxes. Find the dimensions of the cracker.

2014 Kukin MO - Omsk State University 7.5

A right angle is drawn on a sheet of paper with apex $O$. Seven-grader Semyon wants to draw several rays with the same vertex lying inside the angle, so that the sum of all the angles obtained in this case is $400$ degrees. What is the smallest number of rays it can conduct and how should they be positioned?

2014 Kukin MO - Omsk State University 8.4

On the sides $AB$, $BC$, $CD$ and $DA$ of the quadrilateral $ABCD$, the points $K$, $L$, $M$, $N$ are selected, respectively, so that $AK = AN$, $BK = BL$, $CL = CM$, $DM = DN$ and $KLMN$ is a rectangle. Prove that $ABCD$ is a rhombus.

2014 Kukin MO - Omsk State University 9.2

Four circles are located on the plane so that the first one touches the second, the second with the third, the third with the fourth, and the fourth with the first, and all tangencies are external. The tangency points form a rectangle. The radii of some two circles are $1$ and $2$ cm. Find the radii of the other two circles.

2014 Kukin MO - Omsk State University 10.3

Four circles are located on the plane so that the first one touches the second, the second with the third, the third with the fourth, and the fourth with the first, and all touches are external. The tangency points form a square. Does it follow from this that all circles have the same radius?

2014 Kukin MO - Omsk State University 11.2

A spatial quadrilateral is a closed polyline in space without self-intersections. Can all four angle $BAD$, $ABC$, $BCD$, $CDA$ of spatial quadrilateral $ABCD$ be less than one degree?

2014 Kukin MO - Omsk State University 11.3

The math teacher drew a right-angled triangle on the blackboard. Petya, an eleventh grader, assures that he can increase all the sides of this triangle by the same amount, so that again we get a right-angled triangle. And his friend Vasya says that he can reduce all the sides of this triangle by the same amount, so that we will also get a right-angled triangle. Which one should you believe?

2014 Kukin MO - Omsk State University 11.4

The triangle $ABC$ is inscribed in the circle $\omega$. Point $M$ lies on arc $BC$ of this circle, which does not contain point $A$. The tangents to the incircle of triangle $ABC$, drawn from point $M$, intersect the circle $\omega$ at points $N$ and $P$. Moreover, $\angle BAC = \angle NMP$. Prove that triangles $ABC$ and $MNP$ are congruent.

2015 Kukin MO - Omsk State University 8.4

The bisectors of the angles $A$ and $C$ cut the non-isosceles triangle $ABC$ into a quadrilateral and three triangles, and among these three triangles there are two isosceles. Find the angles of the triangle $ABC$.

2015 Kukin MO - Omsk State University 9.4

Angle $B$ of triangle $ABC$ is $45$ degrees. Point $P$ is chosen inside the triangle so that $BP = 6$ cm and $\angle BAP= \angle BCP= 45^o$. Find the area of a quadrilateral $BAPC$.

(Bulgarian Olympiads)

2015 Kukin MO - Omsk State University 10.4

The bisector of angle $A$ of triangle $ABC$ intersects the side $BC$ at point $P$. On the segment $AP,$ an arbitrary point $M$ is selected, and point $N$ is symmetric to point $M$ wrt the midpoint of the segment $BC$. Line $CN$ intersects line $AB$ at point $E$, line $BN$ intersects line $AC$ at point $D$. Prove that $CD = BE$.

(Bulgarian Olympiads)

2015 Kukin MO - Omsk State University 11.3

In triangle $ABC$, point $M$ is the midpoint of side $AB$. The extension of the median $CM$ intersects the circumcircle of triangle $ABC$ at point $P$. Point $Q$ is chosen on the segment $CM$ so that $PM = MQ$. Line $BQ$ intersects side $AC$ at point $R$, such that $CRMB$ is cyclic. Prove that lines $BR$ and $AC$ are perpendicular.

(Bulgarian Olympiads)

2016 Kukin MO - Omsk State University 8.4

In triangle $ABC$, the angle $C$ is $2$ times the angle $B$, $CD$ is the angle bisector. From the midpoint $M$ of the side $BC$ the perpendicular $MH$ is drawn on the segment $CD$. There is a point $K$ on the side $AB$ such that $KMH$ is an equilateral triangle. Prove that the points $M$, $H$ and $A$ are collinear.

2016 Kukin MO - Omsk State University 9.3

In triangle $ABC$, angle C is right, angle $B = 60^o$, $BD$ is an angle bisector. Point $K$ is selected on the segment $AC$ anf point $M$ is selected on the segment $AB$, such that the triangle $KMD$ is equilateral. Find the ratio $MC / AK$ .

(Usov S.V.)

2016 Kukin MO - Omsk State University 10.3

On sides $AB, BC, CD$ and $AD$ of quadrilateral $ABCD$, respectively points $K, L, M, N$ are selected so that $AK / KB = BL / LC = CM / MD = DN / NA$. On the the sides $KL, LM, MN$ and $NK$ are respectively selected points $P, Q, R, S$ so, that $KP / PL = LQ / QM = MR / RN = DS / SA$. Prove that the area a quadrilateral $PQRS$ is not less than a quarter of the area of a quadrilateral $ABCD$.

(Kukina E.G.)

2016 Kukin MO - Omsk State University 10.6

A circle is drawn through vertex $B$ of triangle $ABD$ tangent to sides $AD$ in its midpoint $E$, and intersecting sides $AB$ and $BD$ at points $K$ and $L$, respectively. On the arc $BL$ that does not contain point $K$, point $C$ is selected so that $\angle CKL = \angle BDA$. In this case, the triangle $KCD$ turned out to be equilateral. Find the angle $BAD$.

(Foreign Olympiads)

2016 Kukin MO - Omsk State University 11.3

There is a paper regular tetrahedron. Petya and Vasya take turns doing cuts with a razor blade along the edges of the tetrahedron until one of the possible sweeps will turn out. Petya makes the first move. Petya wants the resulting paper to have the shape of a triangle, and Vasya wants it to have the shape of a parallelogram. Which player has a strategy to get it done?

(Chernyavskaya I.A.)

2017 Kukin MO - Omsk State University 7.5

Three ribbons of the same width are laid as in the picture, forming equilateral triangles. The perimeter of the internal triangle $16$ cm, the perimeter of each quadrangle on crossing the ribbons is $4$ cm. Find the perimeter of the outer triangle. Be sure to justify your answer.

(A. Shapovalov)

2017 Kukin MO - Omsk State University 8.2

The teacher drew a rectangle $ABCD$ on the blackboard. Student Petya divided this rectangle into two rectangles with a straight line parallel to the side $AB$. It turned out that the areas of these parts are $1: 2$, and the perimeters are $3: 5$ (in the same order). Pupil Vasya divided this rectangle into two parts by a straight line parallel to the $BC$ side. The area of the new parts is also $1: 2$. What is the ratio of their perimeters ?

2017 Kukin MO - Omsk State University 8.3

Point $K$ is taken in triangle $ABC$ on side $BC$. $KM$ and $KP$ are the angle bisectors of triangles $AKB$ and $AKC$, respectively. It turned out that the diagonal $MK$ divides the quadrilateral $BMPK$ into two equal triangles. Prove that $M$ is the midpoint of $AB$.

2017 Kukin MO - Omsk State University 8.6

Petya marks four points on the plane so that all of them cannot be crossed out by two parallel straight lines. Vasya chooses two of the lines passing through pairs of points, measures the angle between them and pays Petya an amount equal to the degree measure of the angle. What is the largest amount Petya can guarantee himself?

2017 Kukin MO - Omsk State University 9.4

In an cyclic $21$-gon, all angles are measured in an integer degrees. Prove that the polygon has parallel diagonals.

(Shapovalov A.V.)

2017 Kukin MO - Omsk State University 9.5

Points $A_1, B_1, C_1$ are selected on the sides $BC, CA, AB$ of triangle $ABC$ respectively so that the straight line $A_1C_1$ intersects the straight line $AC$ at the point $K$, located behind beyond $A$. Line $A_1B_1$ intersects line $AB$ in point $M$, located also beyond point $A$. It is known that the areas four triangles $KAC_1, BA_1X_1, CA_1B_1$ and $MAB_1$ are equal. Prove that the straight lines $BC$ and $KM$ are parallel.

(S.V. Usov)

2017 Kukin MO - Omsk State University 10.4

Points $K$ and $M$ are selected on the sides $AB$ and $BC$ of triangle $ABC$ respectively. The segments $AM$ and $SK$ intersect at point $P$. We know that $KB \cdot PC = PM \cdot BC$. Prove that either $BKPM$ or $AKMC$ is cyclic quadrilateral.

(Chernyavskaya I.A., Kukina E.G.)

2018 Kukin MO - Omsk State University 7.3

In triangle $ABC$, the median drawn from vertex $A$, splits it in two triangle of the same perimeter. Is it possible to draw a ray from the same vertex that breaks the original triangle into two such that the perimeter of one is twice the perimeter of the other?

(S. Usov)

2018 Kukin MO - Omsk State University 8.4

In a convex pentagon $ABCDE$, $AB$ is parallel to $DE$, $CD = DE$, $CE$ is perpendicular to $BC$ and $AD$. Prove that the line passing through $A$ parallel to $CD$, the line passing through $B$ parallel to $CE$, and the line passing through $E$ parallel to $BC$, intersect at one point.

2018 Kukin MO - Omsk State University 9.3

Diameters of circles, drawn from the point of their intersection, form a right angle. The distance between the other ends of the same diameters $2.5$ times the common chord of the circles. Find the ratio of the radii of the circles.

(S. Usov)

2018 Kukin MO - Omsk State University 10.2

There are $4$ sticks. Putting aside one at a time, the remaining three sticks create a triangle. All $4$ triangles have equal areas. Do all $4$ sticks have to be the same length?

(A. Shapovalov)

2018 Kukin MO - Omsk State University 10.5

Prove that the line through the midpoint of the side of a triangle and dividing it into two polygons of the same perimeter, is parallel to the bisector of the opposite angle.

(Kharkiv Olympiads)

2019 Kukin MO - Omsk State University 7.3

In an acute-angled triangle $ABC$, the angle $A$ is $40^o$. Point $M$ is selected on the side $AC$ so that both triangles $ABM$ and $BCM$ are isosceles. Find the rest of the angles of triangle $ABC$.

(A. Adelshin)

2019 Kukin MO - Omsk State University 9.4

Isosceles triangles, $AMB,BTC, CKA$, are built on the sides of the triangle $ABC$ in such a way that the circle circumscribed around $ABC$ is inscribed in a isosceles triangle $MTK$ and the angle $AMB$ is $60$ degrees. Find the ratio of the radii of the circles inscribed in the $BTC$ and $CKA$.

(S. Usov)

2019 Kukin MO - Omsk State University 10.6

In the triangle $ABC$, the angle bisector $CK$ is drawn and the point $M$ is selected on the $BC$, such that $KB \cdot PC =BC \cdot PM$, where $P$ is the point of intersection of $AM$ and $CK$. Inside point $X$ is chosen for the angle $APC$ so that $AX = MC$ and the angle $KXC$ is equal to the angle $KPA$ . Prove that either the point $P$ lies on the circumcircle of the triangle $KBM$, or the point of intersection of $CM$ and $AX$ lies on the circumscribed circle triangle $ABM$.

(S. Usov)

2020 Kukin MO - Omsk State University 7.3

In the pentagon $ABCDE$, $AB = BC$, $CD = DE$. Angle bisectors drawn from the certices $B$ and $D$ intersect at the point $O$ lying on the side $AE$. In which ratio does the point $O$ divide the side $AE$ ?

2020 Kukin MO - Omsk State University 7.6

The square was divided into several centrally symmetric polygons that are not rectangles. In each the center of the polygon is marked. The boundaries of the polygons have been erased leaving only the borders of the square and the centers. Is it always possible unambiguously to restore the blurred boundaries?

2020 Kukin MO - Omsk State University 8.2

In pentagon $ABCDE$, each diagonal is parallel to one of the sides. The side $AB$ was extended to the intersection with the extensions of the sides $CD$ and $DE$ , the intersection points were denoted by $X$ and $Y$. Prove that $AX = BY$.

(S. Usov)

2020 Kukin MO - Omsk State University 9.4

In a pentagon, each diagonal is parallel to some side. The extensions of the sides were extended to the intersection, receiving a "big star". How many times the perimeter of the "big star" is greater than the perimeter of the "small star", formed by the diagonals of the pentagon?

(S. Usov)

2020 Kukin MO - Omsk State University 10.4

Circle $C_1$ of radius $3$ and circle $C_2$ of radius $9$ touch at point $B$, $\ell$ is their common tangent, tangent at them at points $E$ and $D$. Circle $C_3$ passes through points $B, E, D$. Prove that the triangle formed by points $B, E$ and the point of intersection of $\ell$ with the tangent drawn at point $B$ to the circle $C_3$, is isosceles.

(S. Usov)

2021 Kukin MO - Omsk State University 8.5

In a convex quadrilateral, the diagonals intersect at right angles. Prove that the midpoints of the segments connecting two of its opposite vertices with midpoints of non-adjacent sides form a rectangle.

2022 Kukin MO - Omsk State University 7.3

Three endless straight furrows were drawn on the field of miracles, which form a triangle. Pinocchio ran to one intersection, measured the angle, ran to another, measured, ran to the third. Then he folded the three angles obtained. And he says that the sum of the measured angles is $140$ degrees. Pierrot knows that the sum of the angles of a triangle cannot equal $140$ degrees, but he claims that he can find one angle! Is Pierrot right? If yes, find that corner. If not, justify your answer. 

(Kukina E.G.)

2022 Kukin MO - Omsk State University 8.3

The diagonal of a parallelogram makes an angle of $36^o$ with its side, which is one third of the total angle. Prove that the ratio of the larger side to the smaller side does not exceed two.

(Kruglova I.A.)

2022 Kukin MO - Omsk State University 9.1

Find the largest value of the smallest angle of a triangle if one of its sides is twice as long as the other.

(Meshcheryakov E.A.)

2022 Kukin MO - Omsk State University 10.2 11.2

A polygon $A_1A_2…A_n$ is given. Points $B_1,B_2,…,B_n$ are marked on sides $A_1A_2$, $A_2A_3$,$…$, $A_nA_1$ such that $\frac{A_1 B_1}{B_1 A_2 }= \frac{A_2 B_2}{B_2 A_3 }=...= \frac{A_n B_n}{ B_n A_1 }$. Further, for each vertex $A_k$, a vector $A_kB_j$ is drawn, and each of the points $A_k$ and $B_j$ is used once. Prove that the sum of the resulting vectors is equal to zero.

(Zadvornov V.S.)

2022 Kukin MO - Omsk State University 10.4 11.4

$S_1$ and $S_2$ are the excircles of the triangle $ABC$, touching the sides $AB$ and $AC$, respectively, and their line of centers is parallel to the bisector of the angle $HCB$, where $H$ is the touchpoint between $S_1$ and the circle inscribed in $ABC$. Find the ratio of the areas of the circle $S_2$ and triangle $ABC$.

(Usov S.V.)

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