Novosibirsk Oral Geometry 2016-21 VII-IX (Russia) 58p

geometry problems from Novosibirsk Oral Olympiad in Geometry for grades 7-9 , by Novosibirsk City Mathematical Circle "Owlet" (Russia) with aops links

Новосибирскую устную олимпиаду по геометрии

collected inside aops here 

inside aops collected, grade VII, VIII, IX

                           2016-17, 2019-21

it did not take place in 2018

2016 Novosibirsk Oral Geometry Olympiad 8-9 p1

In the quadrilateral $ABCD$, angles $B$ and $C$ are equal to $120^o$, $AB = CD = 1$, $CB = 4$. Find the length $AD$.

2016 Novosibirsk Oral Geometry Olympiad 8-9 p2

Bisector of one angle of triangle $ABC$ is equal to the bisector of its external angle at the same vertex (see figure). Find the difference between the other two angles of the triangle.

2016 Novosibirsk Oral Geometry Olympiad 8-9 p3

A square is drawn on a sheet of grid paper on the sides of the cells $ABCD$ with side $8$. Point $E$ is the midpoint of side $BC$, $Q$ is such a point on the diagonal $AC$ such that $AQ: QC = 3: 1$. Find the angle between straight lines $AE$ and $DQ$.

2016 Novosibirsk Oral Geometry Olympiad 8-9 p4

The two angles of the squares are adjacent, and the extension of the diagonals of one square intersect the diagonal of another square at point $O$ (see figure). Prove that $O$ is the midpoint of $AB$.

2016 Novosibirsk Oral Geometry Olympiad 8-9 p5

In the parallelogram $CMNP$ extend the bisectors of angles $MCN$ and $PCN$ and intersect with extensions of sides PN and $MN$ at points $A$ and $B$, respectively. Prove that the bisector of the original angle $C$ of the the parallelogram is perpendicular to $AB$.

2016 Novosibirsk Oral Geometry Olympiad 8-9 p6

An arbitrary point $M$ inside an equilateral triangle $ABC$ was connected to vertices. Prove that on each side the triangle can be selected one point at a time so that the distances between them would be equal to $AM, BM, CM$.

2017 Novosibirsk Oral Geometry Olympiad 8-9 p1

Petya (Петя ) and Vasya (Вася ) live in neighboring houses (see the plan in the figure). Vasya lives in the fourth entrance. It is known that Petya runs to Vasya by the shortest route (it is not necessary walking along the sides of the cells) and it does not matter from which side he runs around his house. Determine in which entrance he lives Petya .

2017 Novosibirsk Oral Geometry Olympiad 8-9 p2

You are given a convex quadrilateral $ABCD$. It is known that $\angle CAD = \angle DBA = 40^o$, $\angle CAB = 60^o$, $\angle CBD = 20^o$. Find the angle $\angle CDB$.

2017 Novosibirsk Oral Geometry Olympiad 8-9 p3

Medians $AA_1, BB_1, CC_1$ and altitudes $AA_2, BB_2, CC_2$ are drawn in triangle $ABC$ . Prove that the length of the broken line $A_1B_2C_1A_2B_1C_2A_1$ is equal to the perimeter of triangle $ABC$.

2017 Novosibirsk Oral Geometry Olympiad 8-9 p4

On grid paper, mark three nodes so that in the triangle they formed, the sum of the two smallest medians equals to half-perimeter.

2017 Novosibirsk Oral Geometry Olympiad 8-9 p5

Point $K$ is marked on the diagonal $AC$ in rectangle $ABCD$ so that $CK = BC$. On the side $BC$, point $M$ is marked so that $KM = CM$. Prove that $AK + BM = CM$.

2017 Novosibirsk Oral Geometry Olympiad 8-9 p6

In trapezoid $ABCD$, diagonal $AC$ is the bisector of angle $A$. Point $K$ is the midpoint of diagonal $AC$. It is known that $DC = DK$. Find the ratio of the bases $AD: BC$.

2017 Novosibirsk Oral Geometry Olympiad 8-9 p7

A car is driving along a straight highway at a speed of $60$ km per hour. Not far from the highway there is a parallel to him a $100$-meter fence. Every second, the passenger of the car measures the angle at which the fence is visible. Prove that the sum of all the angles he measured is less than $1100^o$

it did not take place in 2018

2019 Novosibirsk Oral Geometry Olympiad 7.1

Lyuba, Tanya, Lena and Ira ran across a flat field. At some point it turned out that among the pairwise distances between them there are distances of $1, 2, 3, 4$ and $5$ meters, and there are no other distances. Give an example of how this could be.

2019 Novosibirsk Oral Geometry Olympiad 7.2 8.1

Kikoriki live on the shores of a pond in the form of an equilateral triangle with a side of $600$ m, Krash and Wally live on the same shore, $300$ m from each other. In summer, Dokko to Krash walk $900$ m, and Wally to Rosa - also $900$ m. Prove that in winter, when the pond freezes and it will be possible to walk directly on the ice, Dokko will walk as many meters to Krash as Wally to Rosa.

2019 Novosibirsk Oral Geometry Olympiad 7.3

Equal line segments are marked in triangle $ABC$. Find its angles.

2019 Novosibirsk Oral Geometry Olympiad 7.4

Two squares and an isosceles triangle are positioned as shown in the figure (the up left vertex of the large square lies on the side of the triangle). Prove that points $A, B$ and $C$ are collinear.

2019 Novosibirsk Oral Geometry Olympiad 7.5 8.4 9.4

Given a triangle $ABC$, in which the angle $B$ is three times the angle $C$. On the side $AC$, point $D$ is chosen such that the angle $BDC$ is twice the angle $C$. Prove that $BD + BA = AC$.

2019 Novosibirsk Oral Geometry Olympiad 7.6 8.5

Two turtles, the leader and the slave, are crawling along the plane from point $A$ to point $B$. They crawl in turn: first the leader crawls some distance, then the slave crawls some distance in a straight line towards the leading one. Then the leader crawls somewhere again, after which the slave crawls towards the leader, etc. Finally, they both crawl to $B$. Prove that the slave turtle crawled no more than the leading one.

2019 Novosibirsk Oral Geometry Olympiad 7.7

Cut a square into eight acute-angled triangles.

2019 Novosibirsk Oral Geometry Olympiad 8.1 7.2 

Kikoriki live on the shores of a pond in the form of an equilateral triangle with a side of $600$ m, Krash and Wally live on the same shore, $300$ m from each other. In summer, Dokko to Krash walk $900$ m, and Wally to Rosa - also $900$ m. Prove that in winter, when the pond freezes and it will be possible to walk directly on the ice, Dokko will walk as many meters to Krash as Wally to Rosa.

2019 Novosibirsk Oral Geometry Olympiad 8.2 9.1

The circle is inscribed in a triangle, inscribed in a semicircle. Find the marked angle $a$.

2019 Novosibirsk Oral Geometry Olympiad 8.3

A square sheet of paper $ABCD$ is folded straight in such a way that point $B$ hits to the midpoint of side $CD$. In what ratio does the fold line divide side $BC$?

2019 Novosibirsk Oral Geometry Olympiad 8.4 7.5 9.4

Given a triangle $ABC$, in which the angle $B$ is three times the angle $C$. On the side $AC$, point $D$ is chosen such that the angle $BDC$ is twice the angle $C$. Prove that $BD + BA = AC$.

2019 Novosibirsk Oral Geometry Olympiad 8.5 7.6 

Two turtles, the leader and the slave, are crawling along the plane from point $A$ to point $B$. They crawl in turn: first the leader crawls some distance, then the slave crawls some distance in a straight line towards the leading one. Then the leader crawls somewhere again, after which the slave crawls towards the leader, etc. Finally, they both crawl to $B$. Prove that the slave turtle crawled no more than the leading one.

2019 Novosibirsk Oral Geometry Olympiad 8.6 9.5

Point $A$ is located in this circle of radius $1$. An arbitrary chord is drawn through it, and then a circle of radius $2$ is drawn through the ends of this chord. Prove that all such circles touch some fixed circle, not depending from the initial choice of the chord.

2019 Novosibirsk Oral Geometry Olympiad 8.7

The square was cut into acute -angled triangles. Prove that there are at least eight of them.

2019 Novosibirsk Oral Geometry Olympiad 9.1 8.2

The circle is inscribed in a triangle, inscribed in a semicircle. Find the marked angle $a$.

                                             

2019 Novosibirsk Oral Geometry Olympiad 9.2

An angle bisector $AD$ was drawn in triangle $ABC$. It turned out that the center of the inscribed circle of triangle $ABC$ coincides with the center of the inscribed circle of triangle $ABD$. Find the angles of the original triangle.

2019 Novosibirsk Oral Geometry Olympiad 9.3

The circle touches the square and goes through its two vertices as shown in the figure. Find the area of the square. (Distance in the picture is measured horizontally from the midpoint of the side of the square.)

2019 Novosibirsk Oral Geometry Olympiad 9.4 7.5 8.4 

Given a triangle $ABC$, in which the angle $B$ is three times the angle $C$. On the side $AC$, point $D$ is chosen such that the angle $BDC$ is twice the angle $C$. Prove that $BD + BA = AC$.

2019 Novosibirsk Oral Geometry Olympiad 9.5 8.6

Point $A$ is located in this circle of radius $1$. An arbitrary chord is drawn through it, and then a circle of radius $2$ is drawn through the ends of this chord. Prove that all such circles touch some fixed circle, not depending from the initial choice of the chord.

2019 Novosibirsk Oral Geometry Olympiad 9.6

A square with side $1$ contains a non-self-intersecting polyline of length at least $200$. Prove that there is a straight line parallel to the side of the square that has at least $101$ points in common with this polyline.

2019 Novosibirsk Oral Geometry Olympiad 9.7

Denote $X,Y$ two convex polygons, such that $X$ is contained inside $Y$. Denote $S (X)$, $P (X)$, $S (Y)$, $P (Y)$ the area and perimeter of the first and second polygons, respectively. Prove that $$\frac{S(X)}{P(X)}<2 \frac{S(Y)}{P(Y)}.$$

2020 Novosibirsk Oral Geometry Olympiad 7.1

All twelve points on the circle are at equal distances. The only marked point inside is the center of the circle. Determine which part of the whole circle in the picture is filled in.

2020 Novosibirsk Oral Geometry Olympiad 7.2

It is known that four of these sticks can be assembled into a quadrilateral. Is it always true that you can make a triangle out of three of them?

2020 Novosibirsk Oral Geometry Olympiad 7.3

Cut an arbitrary triangle into $2019$ pieces so that one of them turns out to be a triangle, one is a quadrilateral, ... one is a $2019$-gon and one is a $2020$-gon. Polygons do not have to be convex.

2020 Novosibirsk Oral Geometry Olympiad 7.4

The altitudes $AN$ and $BM$ are drawn in triangle $ABC$. Prove that the perpendicular bisector to the segment $NM$ divides the segment $AB$ in half.

2020 Novosibirsk Oral Geometry Olympiad 7.5 8.4 9.3

Point $P$ is chosen inside triangle $ABC$ so that $\angle APC+\angle ABC=180^o$ and $BC=AP.$ On the side $AB$, a point $K$ is chosen such that $AK = KB + PC$. Prove that $CK \perp AB$.

2020 Novosibirsk Oral Geometry Olympiad 7.6 8.6 9.5

Angle bisectors $AA', BB'$and $CC'$ are drawn in triangle $ABC$ with angle $\angle B= 120^o$. Find $\angle A'B'C'$.

2020 Novosibirsk Oral Geometry Olympiad 7.7

The segments connecting the interior point of a convex non-sided $n$-gon with its vertices divide the $n$-gon into $n$ congruent triangles. For what is the smallest $n$ that is possible?

2020 Novosibirsk Oral Geometry Olympiad 8.1

Three squares of area $4, 9$ and $36$ are inscribed in the triangle as shown in the figure. Find the area of the big triangle.

2020 Novosibirsk Oral Geometry Olympiad 8.2

Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get?

2020 Novosibirsk Oral Geometry Olympiad 8.3

Maria Ivanovna drew on the blackboard a right triangle $ABC$ with a right angle $B$. Three students looked at her and said:

$\bullet$ Yura said: "The hypotenuse of this triangle is $10$ cm."

$\bullet$ Roma said: "The altitude drawn from the vertex $B$ on the side $AC$ is $6$ cm."

$\bullet$ Seva said: "The area of the triangle $ABC$ is $25$ cm$^2$."

Determine which of the students was mistaken if it is known that there is exactly one such person.

2020 Novosibirsk Oral Geometry Olympiad 8.4 7.5 9.3

Point $P$ is chosen inside triangle $ABC$ so that $\angle APC+\angle ABC=180^o$ and $BC=AP.$ On the side $AB$, a point $K$ is chosen such that $AK = KB + PC$. Prove that $CK \perp AB$.

2020 Novosibirsk Oral Geometry Olympiad 8.5

Line $\ell$ is perpendicular to one of the medians of the triangle. The median perpendiculars to the sides of this triangle intersect the line $\ell$ at three points. Prove that one of them is the midpoint of the segment formed by the other two.

2020 Novosibirsk Oral Geometry Olympiad 8.6 7.6 9.5

Angle bisectors $AA', BB'$and $CC'$ are drawn in triangle $ABC$ with angle $\angle B= 120^o$. Find $\angle A'B'C'$.

2020 Novosibirsk Oral Geometry Olympiad 8.7

You are given a quadrilateral $ABCD$. It is known that $\angle BAC = 30^o$, $\angle D = 150^o$ and, in addition, $AB = BD$. Prove that $AC$ is the bisector of angle $C$.

2020 Novosibirsk Oral Geometry Olympiad 9.1

Two semicircles touch the side of the rectangle, each other and the segment drawn in it as in the figure. What part of the whole rectangle is filled?

2020 Novosibirsk Oral Geometry Olympiad 9.2

A $2 \times 2$ square was cut out of a sheet of grid paper. Using only a ruler without divisions and without going beyond the square, divide the diagonal of the square into $6$ equal parts.

2020 Novosibirsk Oral Geometry Olympiad 9.3 7.5 8.4

Point $P$ is chosen inside triangle $ABC$ so that $\angle APC+\angle ABC=180^o$ and $BC=AP.$ On the side $AB$, a point $K$ is chosen such that $AK = KB + PC$. Prove that $CK \perp AB$.

2020 Novosibirsk Oral Geometry Olympiad 9.4

Points $E$ and $F$ are the midpoints of sides $BC$ and $CD$ of square $ABCD$, respectively. Lines $AE$ and $BF$ meet at point $P$. Prove that $\angle PDA = \angle AED$.

2020 Novosibirsk Oral Geometry Olympiad 9.5 7.6 8.6 

Angle bisectors $AA', BB'$and $CC'$ are drawn in triangle $ABC$ with angle $\angle B= 120^o$. Find $\angle A'B'C'$.

2020 Novosibirsk Oral Geometry Olympiad 9.6

In triangle $ABC$, point $M$ is the midpoint of $BC$, $P$ the point of intersection of the tangents at points $B$ and $C$ of the circumscribed circle of $ABC$, $N$ is the midpoint of the segment $MP$. The segment $AN$ meets the circumcircle $ABC$ at the point $Q$. Prove that $\angle PMQ = \angle MAQ$.

2020 Novosibirsk Oral Geometry Olympiad 9.7

The quadrilateral $ABCD$ is known to be inscribed in a circle, and that there is a circle with center on side $AD$ tangent to the other three sides. Prove that $AD = AB + CD$.

2021 Novosibirsk Oral Geometry Olympiad 7.1 8.1

Cut the $9 \times 10$ grid rectangle along the grid lines into several squares so that there are exactly two of them with odd sidelengths.

2021 Novosibirsk Oral Geometry Olympiad 7.2 8.2

The extensions of two opposite sides of the convex quadrilateral intersect and form an angle of $20^o$ , the extensions of the other two sides also intersect and form an angle of $20^o$. It is known that exactly one angle of the quadrilateral is $80^o$. Find all of its other angles.

2021 Novosibirsk Oral Geometry Olympiad 7.3

Prove that in a triangle one of the sides is twice as large as the other if and only if a median and an angle bisector of this triangle are perpendicular

2021 Novosibirsk Oral Geometry Olympiad 7.4

It is known about two triangles that for each of them the sum of the lengths of any two of its sides is equal to the sum of the lengths of any two sides of the other triangle. Are triangles necessarily congruent?

2021 Novosibirsk Oral Geometry Olympiad 7.5

In an acute-angled triangle $ABC$ on the side $AC$, point $P$ is chosen in such a way that $2AP = BC$. Points $X$ and $Y$ are symmetric to $P$ with respect to vertices $A$ and $C$, respectively. It turned out that $BX = BY$. Find $\angle BCA$.

2021 Novosibirsk Oral Geometry Olympiad 7.6 8.6

Inside the equilateral triangle $ABC$, points $P$ and $Q$ are chosen so that the quadrilateral $APQC$ is convex, $AP = PQ = QC$ and $\angle PBQ = 30^o$. Prove that $AQ = BP$.

2021 Novosibirsk Oral Geometry Olympiad 7.7 8.7 9.6

Two congrurent rectangles are located as shown in the figure. Find the area of the shaded part.

                                 

2021 Novosibirsk Oral Geometry Olympiad 8.1 7.1

Cut the $9 \times 10$ grid rectangle along the grid lines into several squares so that there are exactly two of them with odd sidelengths.

2021 Novosibirsk Oral Geometry Olympiad 8.2 7.2

The extensions of two opposite sides of the convex quadrilateral intersect and form an angle of $20^o$ , the extensions of the other two sides also intersect and form an angle of $20^o$. It is known that exactly one angle of the quadrilateral is $80^o$. Find all of its other angles.

2021 Novosibirsk Oral Geometry Olympiad 8.3

Find the angle $BCA$ in the quadrilateral of the figure.

2021 Novosibirsk Oral Geometry Olympiad 8.4

Angle bisectors $AD$ and $BE$ are drawn in triangle $ABC$. It turned out that $DE$ is the bisector of triangle $ADC$. Find the angle $BAC$.

2021 Novosibirsk Oral Geometry Olympiad 8.5

On the legs $AC$ and $BC$ of an isosceles right-angled triangle with a right angle $C$, points $D$ and $E$ are taken, respectively, so that $CD = CE$. Perpendiculars on line $AE$ from points $C$ and $D$ intersect segment $AB$ at points $P$ and $Q$, respectively. Prove that $BP = PQ$.

2021 Novosibirsk Oral Geometry Olympiad 8.6 7.6

Inside the equilateral triangle $ABC$, points $P$ and $Q$ are chosen so that the quadrilateral $APQC$ is convex, $AP = PQ = QC$ and $\angle PBQ = 30^o$. Prove that $AQ = BP$.

2021 Novosibirsk Oral Geometry Olympiad 8.7 7.7 9.6

Two congrurent rectangles are located as shown in the figure. Find the area of the shaded part.

2021 Novosibirsk Oral Geometry Olympiad 9.1

Cut the $19 \times 20$ grid rectangle along the grid lines into several squares so that there are exactly four of them with odd sidelengths.

2021 Novosibirsk Oral Geometry Olympiad 9.2

The robot crawls the meter in a straight line, puts a flag on and turns by an angle $a <180^o$ clockwise. After that, everything is repeated. Prove that all flags are on the same circle.

2021 Novosibirsk Oral Geometry Olympiad 9.3

In triangle $ABC$, side $AB$ is $1$. It is known that one of the angle bisectors of triangle $ABC$ is perpendicular to one of its medians, and some other angle bisector is perpendicular to the other median. What can be the perimeter of triangle $ABC$?

2021 Novosibirsk Oral Geometry Olympiad 9.4

A semicircle of radius $5$ and a quarter of a circle of radius $8$ touch each other and are located inside the square as shown in the figure. Find the length of the part of the common tangent, enclosed in the same square.

2021 Novosibirsk Oral Geometry Olympiad 9.5

The pentagon $ABCDE$ is inscribed in the circle. Line segments $AC$ and $BD$ intersect at point $K$. Line segment $CE$ touches the circumcircle of triangle $ABK$ at point $N$. Find the angle $CNK$ if $\angle ECD = 40^o.$

2021 Novosibirsk Oral Geometry Olympiad 9.6 7.7 8.7 

Two congrurent rectangles are located as shown in the figure. Find the area of the shaded part.

                                                 

2021 Novosibirsk Oral Geometry Olympiad 9.7

A circle concentric with the inscribed circle of $ABC$ intersects the sides of the triangle at six points forming a convex hexagon $A_1A_2B_1B_2C_1C_2$ (points $C_1$ and $C_2$ on the $AB$ side, $A_1$ and $A_2$ on $BC$, $B_1$ and $B_2$ on $AC$). Prove that if line $A_1B_1$ is parallel to the bisector of angle $B$, then line $A_2C_2$ is parallel to the bisector of angle $C$.

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