geometry problems

with aops links in the names

from Vyacheslav Yasinsky Geometry Olympiad (Ukrainian)

in memorial of

2017 Yasinsky Geometry Olympiad VIII-IX p1

Rectangular sheet of paper $ABCD$ is folded as shown in the figure. Find the rato $DK: AB$, given that $C_1$ is the midpoint of $AD$.

2017 Yasinsky Geometry Olympiad VIII-IX p2, advanced p1

2017 Yasinsky Geometry Olympiad VIII-IX p2, advanced p1

2017 Yasinsky Geometry Olympiad VIII-IX p3, advanced p2

Given circle arc, whose center is an inaccessible point. $A$ is a point on this arc (see fig.). How to construct use compass and ruler without divisions, a tangent to given circle arc at point $A$ ?

2017 Yasinsky Geometry Olympiad VIII-IX advanced p3

The two sides of the triangle are $10$ and $15$. Prove that the length of the bisector of the angle between them is less than $12$.

$AH$ is the altitude of the acute triangle $ABC$, $K$ and $L$ are the feet of the perpendiculars, from point $H$ on sides $AB$ and $AC$ respectively. Prove that the angles $BKC$ and $BLC$ are equal.

Point $O$ is the center of circumcircle $\omega$ of the isosceles triangle $ABC$ ($AB = AC$). Bisector of the angle $\angle C$ intersects $\omega$ at the point $W$. Point $Q$ is the center of the circumcircle of the triangle $OWB$. Construct the triangle $ABC$ given the points $Q,W, B$.

The inscribed circle of the triangle $ABC$ touches its sides $AB, BC, CA$, at points $K,N, M$ respectively. It is known that $\angle ANM = \angle CKM$. Prove that the triangle $ABC$ is isosceles.

2019 Yasinsky Geometry Olympiad VIII-IX p1

The sports ground has the shape of a rectangle $ABCD$, with the angle between the diagonals $AC$ and $BD$ is equal to $60^o$ and $AB >BC$. The trainer instructed Andriyka to go first $10$ times on the route $A-C-B-D-A$, and then $15$ more times along the route $A-D-A$. Andriyka performed the task, moving a total of $4.5$ km. What is the distance $AC$?

2019 Yasinsky Geometry Olympiad VIII-IX p2

An isosceles triangle $ABC$ ($AB = AC$) with an incircle of radius $r$ is given. We know that the point $M$ of the intersection of the medians of the triangle $ABC$ lies on this circle. Find the distance from the vertex $A$ to the point of intersection of the bisectrix of the triangle $ABC$.

2019 Yasinsky Geometry Olympiad VIII-IX p3

In the quadrilateral $ABCD$, the angles $B$ and $D$ are right . The diagonal $AC$ forms with the side $AB$ the angle of $40^o$, as well with side $AD$ an angle of $30^o$. Find an acute angle between the diagonals $AC$ and $BD$.

2019 Yasinsky Geometry Olympiad VIII-IX p4

Let $ABC$ be a triangle, $O$ is the center of the circle circumscribed around it, $AD$ the diameter of this circle. It is known that the lines $CO$ and $DB$ are parallel. Prove that the triangle $ABC$ is isosceles.

2019 Yasinsky Geometry Olympiad VIII-IX p5

In the triangle $ABC$, $\angle ABC = \angle ACB = 78^o$. On the sides $AB$ and $AC$, respectively, the points $D$ and $E$ are chosen such that $\angle BCD = 24^o, \angle CBE = 51^o$. Find the measure of angle $\angle BED$.

2019 Yasinsky Geometry Olympiad VIII-IX advanced p1

It is known that in the triangle $ABC$ the distance from the intersection point the bisector to each of the vertices of the triangle does not exceed the diameter of the circle inscribed in this triangle. Find the angles of the triangle $ABC$.

with aops links in the names

from Vyacheslav Yasinsky Geometry Olympiad (Ukrainian)

[Started in 2017]

**Yasinska Α Vyacheslav, (Ясінського В’ячеслава Андрійовича) 1957-2015 .**
grades VIII-IX

2017 Yasinsky Geometry Olympiad VIII-IX p1

Rectangular sheet of paper $ABCD$ is folded as shown in the figure. Find the rato $DK: AB$, given that $C_1$ is the midpoint of $AD$.

2017 Yasinsky Geometry Olympiad VIII-IX p2, advanced p1

Medians $AM$ and $BE$ of a triangle $ABC$ intersect at $O$. The points $O, M, E, C$ lie on one circle. Find the length of $AB$ if $BE = AM =3$.

2017 Yasinsky Geometry Olympiad VIII-IX p3, advanced p2

Given circle arc, whose center is an inaccessible point. $A$ is a point on this arc (see fig.). How to construct use compass and ruler without divisions, a tangent to given circle arc at point $A$ ?

2017 Yasinsky Geometry Olympiad VIII-IX p3, advanced p2

Given circle arc, whose center is an inaccessible point. $A$ is a point on this arc (see fig.). How to construct use compass and ruler without divisions, a tangent to given circle arc at point $A$ ?

Diagonals of trapezium $ABCD$ are mutually perpendicular and the midline of the trapezium is $5$. Find the length of the segment that connects the middles of the bases of the trapezium.

2017 Yasinsky Geometry Olympiad VIII-IX p5, advanced p5

The four points of a circle are in the following order: $A, B, C, D$. Extensions of chord $AB$ beyond point $B$ and of chord $CD$ beyond point $C$ intersect at point $E$, with $\angle AED= 60^o$. If $\angle ABD =3 \angle BAC$ , prove that $AD$ is the diameter of the circle.

2017 Yasinsky Geometry Olympiad VIII-IX advanced p6

Given a trapezoid $ABCD$ with bases $BC$ and $AD$, with $AD=2 BC$. Let $M$ be the midpoint of $AD, E$ be the point of intersection of the sides $AB$ and $CD$, $O$ be the point of intersection of $BM$ and $AC, N$ be the point of intersection of $EO$ and $BC$. In what ratio, point $N$ divides the segment $BC$?

2017 Yasinsky Geometry Olympiad VIII-IX p5, advanced p5

The four points of a circle are in the following order: $A, B, C, D$. Extensions of chord $AB$ beyond point $B$ and of chord $CD$ beyond point $C$ intersect at point $E$, with $\angle AED= 60^o$. If $\angle ABD =3 \angle BAC$ , prove that $AD$ is the diameter of the circle.

2017 Yasinsky Geometry Olympiad VIII-IX advanced p6

Given a trapezoid $ABCD$ with bases $BC$ and $AD$, with $AD=2 BC$. Let $M$ be the midpoint of $AD, E$ be the point of intersection of the sides $AB$ and $CD$, $O$ be the point of intersection of $BM$ and $AC, N$ be the point of intersection of $EO$ and $BC$. In what ratio, point $N$ divides the segment $BC$?

2017 Yasinsky Geometry Olympiad VIII-IX p2, advanced p1

Medians $AM$ and $BE$ of a triangle $ABC$ intersect at $O$. The points $O, M, E, C$ lie on one circle. Find the length of $AB$ if $BE = AM =3$.

2017 Yasinsky Geometry Olympiad VIII-IX p3, advanced p2

Given circle arc, whose center is an inaccessible point. $A$ is a point on this arc (see fig.). How to construct use compass and ruler without divisions, a tangent to given circle arc at point $A$ ?

The two sides of the triangle are $10$ and $15$. Prove that the length of the bisector of the angle between them is less than $12$.

2017 Yasinsky Geometry Olympiad VIII-IX advanced p4

In an isosceles trapezoid, one of the bases is three times larger than the other. Angle at a greater basis is equal to $45^o$. Show how to cut this trapezium into three parts and make a square with them. Justify your answer.

In an isosceles trapezoid, one of the bases is three times larger than the other. Angle at a greater basis is equal to $45^o$. Show how to cut this trapezium into three parts and make a square with them. Justify your answer.

2017 Yasinsky Geometry Olympiad VIII-IX p5, advanced p5

The four points of a circle are in the following order: $A, B, C, D$. Extensions of chord $AB$ beyond point $B$ and of chord $CD$ beyond point $C$ intersect at point $E$, with $\angle AED= 60^o$. If $\angle ABD =3 \angle BAC$ , prove that $AD$ is the diameter of the circle.

2017 Yasinsky Geometry Olympiad VIII-IX advanced p6

Given a trapezoid $ABCD$ with bases $BC$ and $AD$, with $AD=2 BC$. Let $M$ be the midpoint of $AD, E$ be the point of intersection of the sides $AB$ and $CD$, $O$ be the point of intersection of $BM$ and $AC, N$ be the point of intersection of $EO$ and $BC$. In what ratio, point $N$ divides the segment $BC$?

In the isosceles trapezoid with the area of $28$, a circle of radius $2$ is inscribed. Find the length of the side of the trapezoid.

2017 Yasinsky Geometry Olympiad X-XI p2

Prove that if all the edges of the tetrahedron are equal triangles (such a tetrahedron is called equilateral), then its projection on the plane of a face is a triangle.

In the triangle $ABC$ , the angle bisector $AD$ divides the side $BC$ into the ratio $BD: DC = 2: 1$. In what ratio, does the median $CE$ divide this bisector?

2017 Yasinsky Geometry Olympiad X-XI p3, advanced p1

Given circle $\omega$ and point $D$ outside this circle. Find the following points $A, B$ and $C$ on the circle $\omega$ so that the $ABCD$ quadrilateral is convex and has the maximum possible area. Justify your answer.

2017 Yasinsky Geometry Olympiad X-XI advanced p2

In the tetrahedron $DABC, AB=BC, \angle DBC =\angle DBA$. Prove that $AC \perp DB$.

2018 Yasinsky Geometry Olympiad VIII-IX p1, advanced p1

Points $A, B$ and $C$ lie on the same line so that $CA = AB$. Square $ABDE$ and the equilateral triangle $CFA$, are constructed on the same side of line $CB$. Find the acute angle between straight lines $CE$ and $BF$.

2018 Yasinsky Geometry Olympiad VIII-IX p2

Let $ABCD$ be a parallelogram, such that the point $M$ is the midpoint of the side $CD$ and lies on the bisector of the angle $\angle BAD$. Prove that $\angle AMB = 90^o$.

2018 Yasinsky Geometry Olympiad VIII-IX p3, advanced p2

In the triangle $ABC$, $\angle B = 2 \angle C$, $AD$ is altitude, $M$ is the middle of the side $BC$. Prove that $AB = 2DM$.

2018 Yasinsky Geometry Olympiad VIII-IX advanced p3

2018 Yasinsky Geometry Olympiad VIII-IX advanced p4, X-XI advanced p4

Let $I_a$ be the point of the center of an ex-circle of the triangle $ABC$, which touches the side $BC$ . Let $W$ be the point of intersection of the bisector of the angle $\angle A$ of the triangle $ABC$ with the circumcircle of the triangle $ABC$. Perpendicular from the point $W$ on the straight line $AB$, intersects the circumcircle of $ABC$ at the point $P$. Prove, that if the points $B, P, I_a$ lie on the same line, then the triangle $ABC$ is isosceles.

2018 Yasinsky Geometry Olympiad VIII-IX advanced p5

The four points of a circle are in the following order: $A, B, C, D$. Extensions of chord $AB$ beyond point $B$ and of chord $CD$ beyond point $C$ intersect at point $E$, with $\angle AED= 60^o$. If $\angle ABD =3 \angle BAC$ , prove that $AD$ is the diameter of the circle.

2017 Yasinsky Geometry Olympiad VIII-IX advanced p6

Given a trapezoid $ABCD$ with bases $BC$ and $AD$, with $AD=2 BC$. Let $M$ be the midpoint of $AD, E$ be the point of intersection of the sides $AB$ and $CD$, $O$ be the point of intersection of $BM$ and $AC, N$ be the point of intersection of $EO$ and $BC$. In what ratio, point $N$ divides the segment $BC$?

grades X-XI

2017 Yasinsky Geometry Olympiad X-XI p1In the isosceles trapezoid with the area of $28$, a circle of radius $2$ is inscribed. Find the length of the side of the trapezoid.

2017 Yasinsky Geometry Olympiad X-XI p2

Prove that if all the edges of the tetrahedron are equal triangles (such a tetrahedron is called equilateral), then its projection on the plane of a face is a triangle.

2017 Yasinsky Geometry Olympiad X-XI p3, advanced p1

Given circle $\omega$ and point $D$ outside this circle. Find the following points $A, B$ and $C$ on the circle $\omega$ so that the $ABCD$ quadrilateral is convex and has the maximum possible area. Justify your answer.

Given circle $\omega$ and point $D$ outside this circle. Find the following points $A, B$ and $C$ on the circle $\omega$ so that the $ABCD$ quadrilateral is convex and has the maximum possible area. Justify your answer.

2017 Yasinsky Geometry Olympiad X-XI p4

Three points are given on the plane. With the help of compass and ruler construct a straight line in this plane, which will be equidistant from these three points. Explore how many solutions have this construction.

Three points are given on the plane. With the help of compass and ruler construct a straight line in this plane, which will be equidistant from these three points. Explore how many solutions have this construction.

2017 Yasinsky Geometry Olympiad X-XI p5

2017 Yasinsky Geometry Olympiad X-XI p6
ABCD is a rectangle. The segment $MA$ is perpendicular to plane $ABC$ . $MB= 15$ , $MC=24$ , $MD=20$. Find the length of $MA$ .

In the triangle $ABC$ , the angle bisector $AD$ divides the side $BC$ into the ratio $BD: DC = 2: 1$. In what ratio, does the median $CE$ divide this bisector?

2017 Yasinsky Geometry Olympiad X-XI p3, advanced p1

Given circle $\omega$ and point $D$ outside this circle. Find the following points $A, B$ and $C$ on the circle $\omega$ so that the $ABCD$ quadrilateral is convex and has the maximum possible area. Justify your answer.

2017 Yasinsky Geometry Olympiad X-XI advanced p2

In the tetrahedron $DABC, AB=BC, \angle DBC =\angle DBA$. Prove that $AC \perp DB$.

2017 Yasinsky Geometry Olympiad X-XI advanced p3

In a circle, let $AB$ and $BC$ be chords , with $AB =\sqrt3, BC =3\sqrt3, \angle ABC =60^o$. Find the length of the circle chord that divides angle $ \angle ABC$ in half.

In a circle, let $AB$ and $BC$ be chords , with $AB =\sqrt3, BC =3\sqrt3, \angle ABC =60^o$. Find the length of the circle chord that divides angle $ \angle ABC$ in half.

2017 Yasinsky Geometry Olympiad X-XI advanced p4

Median $AM$ and the angle bisector $CD$ of a right triangle $ABC$ ($\angle B=90^o$) intersect at the point $O$. Find the area of the triangle $ABC$ if $CO=9, OD=5$.

Median $AM$ and the angle bisector $CD$ of a right triangle $ABC$ ($\angle B=90^o$) intersect at the point $O$. Find the area of the triangle $ABC$ if $CO=9, OD=5$.

2017 Yasinsky Geometry Olympiad X-XI advanced p5

Find the area of the section of a unit cube $ABCDA_1B_1C_1D_1$, when a plane passes through the midpoints of the edges $AB, AD$ and $CC_1$.

2017 Yasinsky Geometry Olympiad X-XI advanced p6

Find the area of the section of a unit cube $ABCDA_1B_1C_1D_1$, when a plane passes through the midpoints of the edges $AB, AD$ and $CC_1$.

2017 Yasinsky Geometry Olympiad X-XI advanced p6

Given a circle $\omega$ of radius $r$ and a point $A$, which is far from the center of the circle at a distance $d<r.$ Find the geometric locus of vertices $C$ of all possible $ABCD$ rectangles, where points $B$ and $D$ lie on the circle $\omega$ .

2018

grades VIII-IX

Points $A, B$ and $C$ lie on the same line so that $CA = AB$. Square $ABDE$ and the equilateral triangle $CFA$, are constructed on the same side of line $CB$. Find the acute angle between straight lines $CE$ and $BF$.

2018 Yasinsky Geometry Olympiad VIII-IX p2

Let $ABCD$ be a parallelogram, such that the point $M$ is the midpoint of the side $CD$ and lies on the bisector of the angle $\angle BAD$. Prove that $\angle AMB = 90^o$.

2018 Yasinsky Geometry Olympiad VIII-IX p3, advanced p2

In the triangle $ABC$, $\angle B = 2 \angle C$, $AD$ is altitude, $M$ is the middle of the side $BC$. Prove that $AB = 2DM$.

2018 Yasinsky Geometry Olympiad VIII-IX p4

In the quadrilateral $ABCD$, the length of the sides $AB$ and $BC$ is equal to $1$, $\angle B= 100^o , \angle D= 130^o$ . Find the length of $BD$.

In the quadrilateral $ABCD$, the length of the sides $AB$ and $BC$ is equal to $1$, $\angle B= 100^o , \angle D= 130^o$ . Find the length of $BD$.

2018 Yasinsky Geometry Olympiad VIII-IX p5, X-XI p5

In the trapezium $ABCD$ ($AD // BC$), the point $M$ lies on the side of $CD$, with $CM:MD=2:3$, $AB=AD$, $BC:AD=1:3$. Prove that $BD \perp AM$.

In the trapezium $ABCD$ ($AD // BC$), the point $M$ lies on the side of $CD$, with $CM:MD=2:3$, $AB=AD$, $BC:AD=1:3$. Prove that $BD \perp AM$.

2018 Yasinsky Geometry Olympiad VIII-IX p6

In the quadrilateral $ABCD$, the points $E, F$, and $K$ are midpoints of the $AB, BC, AD$ respectively. Known that $KE \perp AB, K F \perp BC$, and the angle $\angle ABC = 118^o$. Find $ \angle ACD$ (in degrees).

2018 Yasinsky Geometry Olympiad VIII-IX p1, advanced p1

Points $A, B$ and $C$ lie on the same line so that $CA = AB$. Square $ABDE$ and the equilateral triangle $CFA$, are constructed on the same side of line $CB$. Find the acute angle between straight lines $CE$ and $BF$.

2018 Yasinsky Geometry Olympiad VIII-IX p3, advanced p2

In the triangle $ABC$, $\angle B = 2 \angle C$, $AD$ is altitude, $M$ is the middle of the side $BC$. Prove that $AB = 2DM$.

In the quadrilateral $ABCD$, the points $E, F$, and $K$ are midpoints of the $AB, BC, AD$ respectively. Known that $KE \perp AB, K F \perp BC$, and the angle $\angle ABC = 118^o$. Find $ \angle ACD$ (in degrees).

2018 Yasinsky Geometry Olympiad VIII-IX p1, advanced p1

Points $A, B$ and $C$ lie on the same line so that $CA = AB$. Square $ABDE$ and the equilateral triangle $CFA$, are constructed on the same side of line $CB$. Find the acute angle between straight lines $CE$ and $BF$.

2018 Yasinsky Geometry Olympiad VIII-IX p3, advanced p2

In the triangle $ABC$, $\angle B = 2 \angle C$, $AD$ is altitude, $M$ is the middle of the side $BC$. Prove that $AB = 2DM$.

2018 Yasinsky Geometry Olympiad VIII-IX advanced p3

Construct triangle $ABC$, given the altitude and the angle bisector both from A, if it is known for the sides of the triangle $ABC$ that $2BC = AB + AC$.

Let $I_a$ be the point of the center of an ex-circle of the triangle $ABC$, which touches the side $BC$ . Let $W$ be the point of intersection of the bisector of the angle $\angle A$ of the triangle $ABC$ with the circumcircle of the triangle $ABC$. Perpendicular from the point $W$ on the straight line $AB$, intersects the circumcircle of $ABC$ at the point $P$. Prove, that if the points $B, P, I_a$ lie on the same line, then the triangle $ABC$ is isosceles.

The point $M$ lies inside the rhombus $ABCD$. It is known that $\angle DAB=110^o$, $\angle AMD=80^o$, $\angle BMC= 100^o$. What can the angle $\angle AMB$ be equal?

2018 Yasinsky Geometry Olympiad VIII-IX advanced p6

Given a triangle $ABC$, in which $AB = BC$. Point $O$ is the center of the circumcircle, point $I$ is the center of the incircle. Point $D$ lies on the side $BC$, such that the lines $DI$ and $AB$ parallel. Prove that the lines $DO$ and $CI$ are perpendicular.

grades X-XI

2018 Yasinsky Geometry Olympiad VIII-IX advanced p6

Given a triangle $ABC$, in which $AB = BC$. Point $O$ is the center of the circumcircle, point $I$ is the center of the incircle. Point $D$ lies on the side $BC$, such that the lines $DI$ and $AB$ parallel. Prove that the lines $DO$ and $CI$ are perpendicular.

(Vyacheslav Yasinsky)

grades X-XI

In the triangle $ABC$, $AD$ is altitude, $M$ is the midpoint of $BC$. It is known that $\angle BAD = \angle DAM = \angle MAC$. Find the values of the angles of the triangle $ABC$

Let $P$ the point of intersection of the diagonals of a convex quadrilateral $ABCD$. It is known that the area of triangles $ABC$, $BCD$ and $DAP$ is equal to $8 cm^2$, $9 cm^2$ and $10 cm^2$. Find the area of the quadrilateral $ABCD$.

In the tetrahedron $SABC$, points $E, F, K, L$ are the midpoints of the sides $SA , BC, AC, SB$ respectively, . The lengths of the segments $EF$ and $KL$ are equal to $11 cm$ and $13 cm$ respectively, and the length of the segment $AB$ equals to $18 cm$. Find the length of the side $SC$ of the tetrahedron.

Let $ABC$ be an acute triangle. A line, parallel to $BC$, intersects sides $AB$ and $AC$ at points $M$ and $P$, respectively. At which placement of points $M$ and $P$, is the radius of the circumcircle of the triangle $BMP$ is the smallest?

In the trapezium $ABCD$ ($AD // BC$), the point $M$ lies on the side of $CD$, with $CM:MD=2:3$, $AB=AD$, $BC:AD=1:3$. Prove that $BD \perp AM$.

In the tetrahedron $SABC$, points $E, F, K, L$ are the midpoints of the sides $SA , BC, AC, SB$ respectively, . The lengths of the segments $EF$ and $KL$ are equal to $11 cm$ and $13 cm$ respectively, and the length of the segment $AB$ equals to $18 cm$. Find the length of the side $SC$ of the tetrahedron.

Let $ABC$ be an acute triangle. A line, parallel to $BC$, intersects sides $AB$ and $AC$ at points $M$ and $P$, respectively. At which placement of points $M$ and $P$, is the radius of the circumcircle of the triangle $BMP$ is the smallest?

Let $I_a$ be the point of the center of an ex-circle of the triangle $ABC$, which touches the side $BC$ . Let $W$ be the point of intersection of the bisector of the angle $\angle A$ of the triangle $ABC$ with the circumcircle of the triangle $ABC$. Perpendicular from the point $W$ on the straight line $AB$, intersects the circumcircle of $ABC$ at the point $P$. Prove, that if the points $B, P, I_a$ lie on the same line, then the triangle $ABC$ is isosceles.

(Vyacheslav Yasinsky)

Let $O$ and $I$ be the centers of the described and inscribed circle the acute-angled triangle $ABC$, respectively. It is known that line $OI$ is parallel to the side $BC$ of this triangle. Line $MI$, where $M$ is the midpoint of $BC$, intersects the altitude $AH$ at the point $T$. Find the length of the segment $IT$, if the radius of the circle inscribed in the triangle $ABC$ is equal to $r$.

2019

grades VIII-IX

2019 Yasinsky Geometry Olympiad VIII-IX p1

The sports ground has the shape of a rectangle $ABCD$, with the angle between the diagonals $AC$ and $BD$ is equal to $60^o$ and $AB >BC$. The trainer instructed Andriyka to go first $10$ times on the route $A-C-B-D-A$, and then $15$ more times along the route $A-D-A$. Andriyka performed the task, moving a total of $4.5$ km. What is the distance $AC$?

2019 Yasinsky Geometry Olympiad VIII-IX p2

An isosceles triangle $ABC$ ($AB = AC$) with an incircle of radius $r$ is given. We know that the point $M$ of the intersection of the medians of the triangle $ABC$ lies on this circle. Find the distance from the vertex $A$ to the point of intersection of the bisectrix of the triangle $ABC$.

In the quadrilateral $ABCD$, the angles $B$ and $D$ are right . The diagonal $AC$ forms with the side $AB$ the angle of $40^o$, as well with side $AD$ an angle of $30^o$. Find an acute angle between the diagonals $AC$ and $BD$.

2019 Yasinsky Geometry Olympiad VIII-IX p4

Let $ABC$ be a triangle, $O$ is the center of the circle circumscribed around it, $AD$ the diameter of this circle. It is known that the lines $CO$ and $DB$ are parallel. Prove that the triangle $ABC$ is isosceles.

In the triangle $ABC$, $\angle ABC = \angle ACB = 78^o$. On the sides $AB$ and $AC$, respectively, the points $D$ and $E$ are chosen such that $\angle BCD = 24^o, \angle CBE = 51^o$. Find the measure of angle $\angle BED$.

The board features a triangle $ABC$, its center of the circle circumscribed is the point $O$, the midpoint of the side $BC$ is the point $F$, and also some point $K$ on side AC (see fig.). Master knowing that $\angle BAC$ of this triangle is equal to the sharp angle $\alpha$ has separately drawn an angle equal to $\alpha$. After this teacher wiped the board, leaving only the points $O, F, K$ and the angle $\alpha$. Is it possible with a compass and a ruler to construct the triangle $ABC$ ? Justify the answer.

2019 Yasinsky Geometry Olympiad VIII-IX advanced p1

It is known that in the triangle $ABC$ the distance from the intersection point the bisector to each of the vertices of the triangle does not exceed the diameter of the circle inscribed in this triangle. Find the angles of the triangle $ABC$.

2019 Yasinsky Geometry Olympiad VIII-IX advanced p2

A scalene triangle $ABC$ is given. It is known that $I$ is the center of the inscribed circle in this triangle, $D, E, F$ points are the touchpoints of this circle with the sides $AB, BC, CA$, respectively. Let $P$ be the intersection point of lines $DE$ and $AI$. Prove that $CP \perp AI$.

A scalene triangle $ABC$ is given. It is known that $I$ is the center of the inscribed circle in this triangle, $D, E, F$ points are the touchpoints of this circle with the sides $AB, BC, CA$, respectively. Let $P$ be the intersection point of lines $DE$ and $AI$. Prove that $CP \perp AI$.

2019 Yasinsky Geometry Olympiad VIII-IX advanced p3

Let $ABCD$ be an inscribed quadrilateral whose diagonals are connected internally. are perpendicular to each other and intersect at the point $P$. Prove that the line connecting the midpoints of the opposite sides of the quadrilateral $ABCD$ bisects the lines $OP$ ($O$ is the center of the circle circumscribed around quadrilateral $ABCD$).

Let $ABCD$ be an inscribed quadrilateral whose diagonals are connected internally. are perpendicular to each other and intersect at the point $P$. Prove that the line connecting the midpoints of the opposite sides of the quadrilateral $ABCD$ bisects the lines $OP$ ($O$ is the center of the circle circumscribed around quadrilateral $ABCD$).

2019 Yasinsky Geometry Olympiad VIII-IX advanced p4, X-XI advanced p4

In the triangle $ABC$, the side $BC$ is equal to $a$. Point $F$ is midpoint of $AB$, $I$ is the point of intersection of the bisectors of triangle $ABC$. It turned out that $\angle AIF = \angle ACB$. Find the perimeter of the triangle $ABC$.

In the triangle $ABC$, the side $BC$ is equal to $a$. Point $F$ is midpoint of $AB$, $I$ is the point of intersection of the bisectors of triangle $ABC$. It turned out that $\angle AIF = \angle ACB$. Find the perimeter of the triangle $ABC$.

2019 Yasinsky Geometry Olympiad VIII-IX advanced p5 (half part if grade X-XI p6)

In the triangle $ABC$ it is known that $BC = 5, AC - AB = 3$. Prove that $r <2 <r_a$ .

(here $r$ is the radius of the circle inscribed in the triangle $ABC$,

$r_a$ is the radius of an exscribed circle that touches the sides of $BC$).

In the triangle $ABC$ it is known that $BC = 5, AC - AB = 3$. Prove that $r <2 <r_a$ .

(here $r$ is the radius of the circle inscribed in the triangle $ABC$,

$r_a$ is the radius of an exscribed circle that touches the sides of $BC$).

2019 Yasinsky Geometry Olympiad VIII-IX advanced p6

In an acute triangle $ABC$ , the bisector of angle $\angle A$ intersects the circumscribed circle of the triangle $ABC$ at the point $W$. From point $W$ , a parallel is drawn to the side $AB$, which intersects this circle at the point $F \ne W$. Describe the construction of the triangle $ABC$, if given are the segments $FA$ , $FW$ and $\angle FAC$.

In an acute triangle $ABC$ , the bisector of angle $\angle A$ intersects the circumscribed circle of the triangle $ABC$ at the point $W$. From point $W$ , a parallel is drawn to the side $AB$, which intersects this circle at the point $F \ne W$. Describe the construction of the triangle $ABC$, if given are the segments $FA$ , $FW$ and $\angle FAC$.

grades X-XI

A circle with center at the origin and radius $5$ intersects the abscissa in points $A$ and $B$. Let $P$ a point lying on the line $x = 11$, and the point $Q$ is the point of intersection of $AP$ with this circle. We know what is the $Q$ point is the midpoint of the $AP$. Find the coordinates of the point $P$.

(here $r$ is the radius of the circle inscribed in the triangle $ABC$).

Given the equilateral triangle $ABC$. It is known that the radius of the inscribed circle is in this triangle is equal to $1$. The rectangle $ABDE$ is such that point $C$ belongs to its side $DE$. Find the radius of the circle circumscribed around the rectangle $ABDE$.

Let $ABCDEF$ be the regular hexagon. It is known that the area of the triangle $ACD$ is equal to $8$. Find the hexagonal area of $ABCDEF$.

Find the angles of the cyclic quadrilateral if you know that each of its diagonals is a bisector of one angle and a trisector of the opposite one (the trisector of the angle is one of the two rays that lie in the middle of the vertex and divide it into three equal parts).

On the sides of the right triangle, outside are constructed regular nonagons, which are built on one of the catheti and on the hypotenuse, with areas equal to $1602$ $cm^2$ and $2019$ $cm^2$, respectively. What is the area of the nonagon that is constructed on the other cathetus of this triangle?

2019 Yasinsky Geometry Olympiad X-XI p6 (this is half part if grade VIII-IX advanced p5)

In the triangle $ABC$ it is known that $BC = 5, AC - AB = 3$. Prove that $r <2$ .(here $r$ is the radius of the circle inscribed in the triangle $ABC$).

The circle $x^2 + y^2 = 25$ intersects the abscissa in points $A$ and $B$. Let $P$ be a point that lies on the line $x = 11$, $C$ is the intersection point of this line with the $Ox$ axis, and the point $Q$ is the intersection point of $AP$ with the given circle. It turned out that the area of the triangle $AQB$ is four times smaller than the area of the triangle $APC$. Find the coordinates of $Q$.

The base of the quadrilateral pyramid $SABCD$ lies the $ABCD$ rectangle with the sides $AB = 1$ and $AD = 10$. The edge $SA$ of the pyramid is perpendicular to the base, $SA = 4$. On the edge of $AD$, find a point $M$ such that the perimeter of the triangle of $SMC$ was minimal.

Two circles $\omega_1$ and $\omega_2$ are tangent externally at the point $P$. Through the point $A$ of the circle $\omega_1$ is drawn a tangent to this circle, which intersects the circle $\omega_2$ at points $B$ and $C$ (see figure). Line $CP$ intersects again the circle $\omega_1$ to $D$. Prove that the PA is a bisector of the angle $DPB$ .

In the triangle $ABC$, the side $BC$ is equal to $a$. Point $F$ is midpoint of $AB$, $I$ is the point of intersection of the bisectors of triangle $ABC$. It turned out that $\angle AIF = \angle ACB$. Find the perimeter of the triangle $ABC$.

In a right triangle $ABC$ with a hypotenuse $AB$, the angle $A$ is greater than the angle $B$. Point $N$ lies on the hypotenuse $AB$ , such that $BN = AC$. Construct this triangle $ABC$ given the point $N$, point $F$ on the side $AC$ and a straight line $\ell$ containing the bisector of the angle $A$ of the triangle $ABC$.

The $ABC$ triangle is given, point $I_a$ is the center of an exscribed circle touching the side $BC$ , the point $M$ is the midpoint of the side $BC$, the point $W$ is the intersection point of the bisector of the angle $A$ of the triangle $ABC$ with the circumscribed circle around him. Prove that the area of the triangle $I_aBC$ is calculated by the formula $S_{ (I_aBC)} = MW \cdot p$, where $p$ is the semiperimeter of the triangle $ABC$.

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