geometry problems
with aops links in the names
from Vyacheslav Yasinsky Geometry Olympiad (Ukrainian)
in memorial of Yasinska Α Vyacheslav, (Ясінського В’ячеслава Андрійовича) 1957-2015 .
2017
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 $12$.
with aops links in the names
from Vyacheslav Yasinsky Geometry Olympiad (Ukrainian)
[Started in 2017]
grades VIII-IX collected inside aops here
grades VIII-IX advanced collected inside aops here
2022 inside aops here
2017
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 $AD$.
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 p6, 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
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 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 p6, 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 $2BC = AB + AC$.
(Alexey Karlyuchenko)
2018 Yasinsky Geometry Olympiad VIII-IX advanced p4, X-XI advanced p4Let $I_a$ 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.
(Mykola Moroz)
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?
(Andrey Mostovy)
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.
(Mykola Moroz)
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.
(Vyacheslav Yasinsky)
Let $O$ and $I$ be the centers of the circumscribed 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 $r$.
(Grigory Filippovsky)
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 $ABC$.
(Grigory Filippovsky)
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.
(Andrey Mostovy)
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.
(Grigory Filippovsky)
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 $ABC$.
(Grigory Filippovsky)
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 $CP \perp AI$.
(Vtalsh Winds)
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 $ABCD$).
(Alexander Dunyak)
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 $ABC$.
(Grigory Filippovsky)
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 $r <2 <r_a$ .
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 $BC$).
(Mykola Moroz)
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 $\angle FAC$.
(Andrey Mostovy)
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).
(Vyacheslav Yasinsky)
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?
(Vladislav Kirilyuk)
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 $ABC$).
(Mykola Moroz)
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 $ABC$.
(Grigory Filippovsky)
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 $ABC$.
(Grigory Filippovsky)
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 $ABC$.
(Mykola Moroz)
2020
grades VIII-IX
In the rectangle $ABCD$, $AB = 2BC$. An equilateral triangle $ABE$ is constructed on the side $AB$ of the rectangle so that its sides $AE$ and $BE$ intersect the segment $CD$. Point $M$ is the midpoint of $BE$. Find the $\angle MCD$.
It is known that the angles of the triangle $ABC$ are $1: 3: 5$. Find the angle between the bisector of the largest angle of the triangle and the line containing the altitude drawn to the smallest side of the triangle.
There is a ruler and a "rusty" compass, with which you can construct a circle of radius $R$. The point $K$ is from the line $\ell$ at a distance greater than $R$. How to use this ruler and this compass to draw a line passing through the point $K$ and perpendicular to line $\ell$?
(Misha Sidorenko, Katya Sidorenko, Rodion Osokin)
The median $AM$ is drawn in the triangle $ABC$ ($AB \ne AC$). The point $P$ is the foot of the perpendicular drawn on the segment $AM$ from the point $B$. On the segment $AM$ we chose such a point $Q$ that $AQ = 2PM$. Prove that $\angle CQM = \angle BAM$.
It is known that a circle can be inscribed in the quadrilateral $ABCD$, in addition $\angle A = \angle C$. Prove that $AB = BC$, $CD = DA$.
(Olena Artemchuk)
Let $ABCD$ be a square, point $E$ be the midpoint of the side $BC$. The point $F$ belongs to the side $AB$, and $DE \perp EF$. The point $G$ lies inside the square, and $GF = FE$ and $GF \perp FE$. Prove that:
a) $DE$ is the bisector of the $\angle FDC$
b) $FG$ is the bisector of the $\angle AFD$
c) the point $G$ is the center of the circle inscribed in the triangle $ADF$.
(Ercole Suppa, Italy)
Given a right triangle $ABC$, the point $M$ is the midpoint of the hypotenuse $AB$. A circle is circumscribed around the triangle $BCM$, which intersects the segment $AC$ at a point $Q$ other than $C$. It turned out that the segment $QA$ is twice as large as the side BC. Find the acute angles of triangle $ABC$.
(Mykola Moroz)
An equilateral triangle $BDE$ is constructed on the diagonal $BD$ of the square $ABCD$, and the point $C$ is located inside the triangle $BDE$. Let $M$ be the midpoint of $BE$. Find the angle between the lines $MC$ and $DE$.
(Dmitry Shvetsov)
Point $M$ is the midpoint of the side $CD$ of the trapezoid $ABCD$, point $K$ is the foot of the perpendicular drawn from point $M$ to the side $AB$. Give that $3BK \le AK$. Prove that $BC + AD\ge 2BM$.
Let $BB_1$ and $CC_1$ be the altitudes of the acute-angled triangle ABC. From the point $B_1$ the perpendiculars $B_1E$ and $B_1F$ are drawn on the sides $AB$ and $BC$ of the triangle, respectively, and from the point $C_1$ the perpendiculars $C_1 K$ and $C_1L$ on the sides $AC$ and $BC$, respectively. It turned out that the lines $EF$ and $KL$ are perpendicular. Find the measure of the angle $A$ of the triangle $ABC$.
(Alexander Dunyak)
Let $AL$ be the bisector of triangle $ABC$. Circle $\omega_1$ is circumscribed around triangle $ABL$. Tangent to $\omega_1$ at point $B$ intersects the extension of $AL$ at point $K$. The circle $\omega_2$ described around the triangle $CKL$ intersects $\omega_1$ a second time at point $Q$, with $Q$ lying on the side $AC$. Find the value of the angle $ABC$.
(Vladislav Radomsky)
In the triangle $ABC$ the altitude $BD$ and $CT$ are drawn, they intersect at the point $H$. The point $Q$ is the foot of the perpendicular dropped from the point $H$ on the bisector of the angle $A$. Prove that the bisector of the external angle $A$ of the triangle $ABC$, the bisector of the angle $BHC$ and the line $QM$, where $M$ is the midpoint of the segment $DT$, intersect at one point.
(Matvsh Kursky)
grades X-XI
The square $ABCD$ is divided into $8$ equal right triangles and the square $KLMN$, as shown in the figure. Find the area of the square $ABCD$ if $KL = 5, PS = 8$.
Let $ABCD$ be a square, point $E$ be the midpoint of the side $BC$. On the side $AB$ mark a point $F$ such that $FE \perp DE$. Prove that $AF + BE = DF$.
(Ercole Suppa, Italy)
A trapezoid $ABCD$ with bases $BC$ and $AD$ is given. The points $K$ and $L$ are chosen on the sides $AB$ and $CD$, respectively, so that $KL \parallel AD$. It turned out that the areas of the quadrilaterals $AKLD$ and $KBCL$ are equal. Find the length $KL$ if $BC = 3, AD = 5$.
In an isosceles trapezoid $ABCD$, the base $AB$ is twice as large as the base $CD$. Point $M$ is the midpoint of $AB$. It is known that the center of the circle inscribed in the triangle $MCB$ lies on the circle circumscribed around the triangle $MDC$. Find the angle $MBC$.
It is known that a circle can be inscribed in the quadrilateral $ABCD$, in addition $\angle A = \angle C$. Prove that $AB = BC$, $CD = DA$.
(Olena Artemchuk)
A cube whose edge is $1$ is intersected by a plane that does not pass through any of its vertices, and its edges intersect only at points that are the midpoints of these edges. Find the area of the formed section. Consider all possible cases.
(Alexander Shkolny)
Given an acute triangle $ABC$. A circle inscribed in a triangle $ABC$ with center at point $I$ touches the sides $AB, BC$ at points $C_1$ and $A_1$, respectively. The lines $A_1C_1$ and $AC$ intersect at the point $Q$. Prove that the circles circumscribed around the triangles $AIC$ and $A_1CQ$ are tangent.
(Dmitry Shvetsov)
On the midline $MN$ of the trapezoid $ABCD$ ($AD\parallel BC$) the points $F$ and $G$ are chosen so that $\angle ABF =\angle CBG$. Prove that then $\angle BAF = \angle DAG$.
(Dmitry Prokopenko)
The segments $BF$ and $CN$ are the altitudes in the acute-angled triangle $ABC$. The line $OI$, which connects the centers of the circumscribed and inscribed circles of triangle $ABC$, is parallel to the line $FN$. Find the length of the al;titude $AK$ in the triangle $ABC$ if the radii of its circumscribed and inscribed circles are $R$ and $r$, respectively.
(Grigory Filippovsky)
The altitudes of the acute-angled triangle $ABC$ intersect at the point $H$. On the segments $BH$ and $CH$, the points $B_1$ and $C_1$ are marked, respectively, so that $B_1C_1 \parallel BC$. It turned out that the center of the circle $\omega$ circumscribed around the triangle $B_1HC_1$ lies on the line $BC$. Prove that the circle $\Gamma$, which is circumscribed around the triangle $ABC$, is tangent to the circle $\omega$ .
It is known about the triangle $ABC$ that $3 BC = CA + AB$. Let the $A$-symmedian of triangle $ABC$ intersect the circumcircle of triangle $ABC$ at point $D$. Prove that $\frac{1}{BD}+ \frac{1}{CD}= \frac{6}{AD}$.
(Ercole Suppa, Italy)
In an isosceles triangle $ABC, I$ is the center of the inscribed circle, $M_1$ is the midpoint of the side $BC, K_2, K_3$ are the points of contact of the inscribed circle of the triangle with segments $AC$ and $AB$, respectively. The point $P$ lies on the circumcircle of the triangle $BCI$, and the angle $M_1PI$ is right. Prove that the lines $BC, PI, K_2K_3$ intersect at one point.
(Mikhail Plotnikov)
2021
The quadrilateral $ABCD$ is known to have $BC = CD = AC$, and the angle $\angle ABC= 70^o$.
Calculate the degree measure of the angle $\angle ADB$.
(Alexey Panasenko)
Given a rectangle $ABCD$, which is located on the line $\ell$. They want to "invert" it by first rotating
around the vertex $D$, and then rotating around the vertex $C$ (see figure). What is the length of the
curve along which vertex $A$ moves at such movement, if $AB = 30$ cm, $BC = 40$ cm?
(Alexey Panasenko)
The segments $AC$ and $BD$ are perpendicular, and $AC$ is twice as large as $BD$ and intersects
$BD$ in it in the midpoint. Find the value of the angle $BAD$, if we know that $\angle CAD = \angle CDB$.
(Gregory Filippovsky)
2021 Yasinsky Geometry Olympiad VIII-IX p4, VIII-IX advanced p1$K$ is an arbitrary point inside the acute-angled triangle $ABC$, in which $\angle A = 30^o$. $F$ and
$N$ are the points of intersection of the medians in the triangles $AKC$ and $AKB$, respectively .
It is known that $FN = q$. Find the radius of the circle circumscribed around the triangle $ABC$.
(Grigory Filippovsky)
Construct an equilateral trapezoid given the height and the midine, if it is known that the midine is
divided by diagonals into three equal parts.
(Grigory Filippovsky)
Given a quadrilateral $ABCD$, around which you can circumscribe a circle. The perpendicular
bisectors of sides $AD$ and $CD$ intersect at point $Q$ and intersect sides $BC$ and $AB$ at points
$P$ and $K$ resepctively. It turned out that the points $K, B, P, Q$ lie on the same circle. Prove that the
points $A, Q, C$ lie on one line.
(Olena Artemchuk)
$K$ is an arbitrary point inside the acute-angled triangle $ABC$, in which $\angle A = 30^o$. $F$ and
$N$ are the points of intersection of the medians in the triangles $AKC$ and $AKB$, respectively .
It is known that $FN = q$. Find the radius of the circle circumscribed around the triangle $ABC$.
(Grigory Filippovsky)
Given a quadrilateral $ABCD$, around which you can circumscribe a circle. The perpendicular
bisectors of sides $AD$ and $CD$ intersect at point $Q$ and intersect sides $BC$ and $AB$ at points
$P$ and $K$ resepctively. It turned out that the points $K, B, P, Q$ lie on the same circle. Prove that the
points $A, Q, C$ lie on one line.
(Olena Artemchuk)
Prove that in triangle $ABC$, the foot of the altitude $AH$, the point of tangency of the inscribed
circle with side $BC$ and projections of point $A$ on the bisectors $\angle B$ and $\angle C$ of the
triangle lie on one circle.
(Dmitry Prokopenko)
Given an acute triangle ABC, in which $\angle BAC = 60^o$. On the sides $AC$ and $AB$ take the
points $T$ and $Q$, respectively, such that $CT = TQ = QB$. Prove that the center of the inscribed
circle of triangle $ATQ$ lies on the side $BC$.
(Dmitry Shvetsov)
A circle is circumscribed around an isosceles triangle $ABC$ with base $BC$. The bisector of the
angle $C$ and the bisector of the angles $A$ intersect the circle at the points $E$ and $D$, respectively,
and the segment $DE$ intersects the sides $BC$ and $AB$ at the points $P$ and $Q$, respectively.
Reconstruct $\vartriangle ABC$ given points $D, P, Q$, if it is known in which half-plane relative to
the line $DQ$ lies the vertex $A$.
2021 Yasinsky Geometry Olympiad VIII-IX advanced p6
In the circle $\omega$, we draw a chord $BC$, which is not a diameter. Point $A$ moves in a circle
$\omega$. $H$ is the orthocenter triangle $ABC$. Prove that for any location of point $A$, a circle
constructed on $AH$ as on diameter, touches two fixed circles $\omega_1$ and $\omega_2$.
(Dmitry Prokopenko)
A regular dodecagon $A_1A_2...A_{12}$ is inscribed in a circle with a diameter of $20$ cm .
Calculate the perimeter of the pentagon $A_1A_3A_6A_8A_{11}$.
(Alexey Panasenko)
In the triangle $ABC$, it is known that $AB = BC = 20$ cm, and $AC = 24$ cm. The point $M$
lies on the side $BC$ and is equidistant from sides $AB$ and $AC$. Find this distance.
(Alexander Shkolny)
Given a rectangular parallelepiped $ABCDA_1B_1C_1D_1$, which has $AD= DC = 3\sqrt2$ cm,
and $DD_1 = 8$ cm. Through the diagonal $B_1D$ of the parallelepipedm parallel to line $A_1C_1$
drawn the plane $\gamma$.
a) Draw a section of a parallelepiped with plane $\gamma$.
b) Justify what geometric figure is this section, and find its area.
(Alexander Shkolny)
Let $BF$ and $CN$ be the altitudes of the acute triangle $ABC$. Bisectors the angles $ACN$ and
$ABF$ intersect at the point $T$. Find the radius of the circle circumscribed around the triangle $FTN$,
if it is known that $BC = a$.
(Grigory Filippovsky)
Circle $\omega$ is inscribed in the $\vartriangle ABC$, with center $I$. Using only a ruler, divide
segment $AI$ in half.
(Grigory Filippovsky)
Three lines were drawn through the point $X$ in space. These lines crossed some sphere at six points.
It turned out that the distances from point $X$ to some five of them are equal to $2$ cm, $3$ cm,
$4$ cm, $5$ cm, $6$ cm. What can be the distance from point $X$ to the sixth point?
(Alexey Panasenko)
Let $BF$ and $CN$ be the altitudes of the acute triangle $ABC$. Bisectors the angles $ACN$ and
$ABF$ intersect at the point $T$. Find the radius of the circle circumscribed around the triangle $FTN$,
if it is known that $BC = a$.
(Grigory Filippovsky)
In the quadrilateral $ABCD$ it is known that $\angle A = 90^o$, $\angle C = 45^o$ . Diagonals $AC$
and $BD$ intersect at point $F$, and $BC = CF$, and the diagonal $AC$ is the bisector of angle $A$.
Determine the other two angles of the quadrilateral $ABCD$.
(Maria Rozhkova)
In the triangle $ABC$, $h_a, h_b, h_c$ are the altitudes and $p$ is its half-perimeter. Compare $p^2$
with $h_ah_b + h_bh_c + h_ch_a$.
(Gregory Filippovsky)
In triangle $ABC$, the point $H$ is the orthocenter. A circle centered at point $H$ and with radius
$AH$ intersects the lines $AB$ and $AC$ at points $E$ and $D$, respectively. The point $X$ is the
symmetric of the point $A$ with respect to the line $BC$ . Prove that $XH$ is the bisector of the angle
$DXE$.
(Matthew of Kursk)
2021 Yasinsky Geometry Olympiad X-XI advanced p5In triangle $ABC$, point $I$ is the center of the inscribed circle. $AT$ is a segment tangent to the
circle circumscribed around the triangle $BIC$ . On the ray $AB$ beyond the point$ B$ and on the
ray $AC$ beyond the point $C$, we draw the segments $BD$ and $CE$, respectively, such that
$BD = CE = AT$. Let the point $F$ be such that $ABFC$ is a parallelogram. Prove that points $D, E$
and $F$ lie on the same line.
(Dmitry Prokopenko)
In an acute-angled triangle $ABC$, point $I$ is the center of the inscribed circle, point $T$ is the
midpoint of the arc $ABC$ of the circumcircle of triangle $ABC$. It turned out that $\angle AIT = 90^o. Prove that $AB + AC = 3BC$.
(Matthew of Kursk)
Official page: http://amnm.vspu.edu.ua/olymp/
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