2019 Novosibirsk Oral Olympiad in Geometry

Part of Novosibirsk Oral Olympiad in Geometry

Subcontests

(7)

2 turtles are crawling from point A to B (2019 Novosibirsk Oral Geo Oly 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.

circles tangent to a fixed circle (2019 Novosibirsk Oral Geo Oly 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.

polyline of length 200 in a square of side 1 (2019 Novosibirsk Oral Geo Oly 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.

incircle of triangle inscribed in semicircle (2019 Novosibirsk Oral Geo Oly 8.2)

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

a

. https://cdn.artofproblemsolving.com/attachments/8/e/334c8662377155086e9211da3589145f460b52.png

walks around or across an equilateral (2019 Novosibirsk Oral Geo Oly 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.
[url=https://en.wikipedia.org/wiki/Kikoriki]about Kikoriki/GoGoRiki / Smeshariki

circumcenter of ABC = incenter of ABD(2019 Novosibirsk Oral Geo Oly 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.

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)}.

BD + BA = AC if <B=3<C, <BDC=2<C (2019 Novosibirsk Oral Geo Oly 7.5)

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

collinear, 2 squares and isosceles (2019 Novosibirsk Oral Geo Oly 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. https://cdn.artofproblemsolving.com/attachments/d/c/03515e40f74ced1f8243c11b3e610ef92137ac.png

angles wanted, equal segments given (2019 Novosibirsk Oral Geo Oly 7.3)

Equal line segments are marked in triangle

ABC

. Find its angles. https://cdn.artofproblemsolving.com/attachments/0/2/bcb756bba15ba57013f1b6c4cbe9cc74171543.png

ratio wanted, paper folding a square (2019 Novosibirsk Oral Geo Oly 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

area wanted, circle tangent to a square (2019 Novosibirsk Oral Geo Oly 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.) https://cdn.artofproblemsolving.com/attachments/7/5/ab4b5f3f4fb4b70013e6226ce5189f3dc2e5be.png

4 points having only distances 1,2,3,4,5 (2019 Novosibirsk Oral Geo Oly 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 Olympiad in Geometry

Subcontests

2 turtles are crawling from point A to B (2019 Novosibirsk Oral Geo Oly 7.6)

circles tangent to a fixed circle (2019 Novosibirsk Oral Geo Oly 8.6)

polyline of length 200 in a square of side 1 (2019 Novosibirsk Oral Geo Oly 9.6)

incircle of triangle inscribed in semicircle (2019 Novosibirsk Oral Geo Oly 8.2)

walks around or across an equilateral (2019 Novosibirsk Oral Geo Oly 7.2)

circumcenter of ABC = incenter of ABD(2019 Novosibirsk Oral Geo Oly 9.2)

cut a square into 8 acute triangles (2019 Novosibirsk Oral Geo Oly 7.7)

&gt;=8 acute triangles from a square (2019 Novosibirsk Oral Geo Oly 8.7)

S(X)/P(X)&lt;2 S(Y)/P(Y), convex polygons (2019 Novosibirsk Oral Geo Oly 9.7)

BD + BA = AC if &lt;B=3&lt;C, &lt;BDC=2&lt;C (2019 Novosibirsk Oral Geo Oly 7.5)

collinear, 2 squares and isosceles (2019 Novosibirsk Oral Geo Oly 7.4)

angles wanted, equal segments given (2019 Novosibirsk Oral Geo Oly 7.3)

ratio wanted, paper folding a square (2019 Novosibirsk Oral Geo Oly 8.3)

area wanted, circle tangent to a square (2019 Novosibirsk Oral Geo Oly 9.3)

4 points having only distances 1,2,3,4,5 (2019 Novosibirsk Oral Geo Oly 7.1)

>=8 acute triangles from a square (2019 Novosibirsk Oral Geo Oly 8.7)

S(X)/P(X)<2 S(Y)/P(Y), convex polygons (2019 Novosibirsk Oral Geo Oly 9.7)

BD + BA = AC if <B=3<C, <BDC=2<C (2019 Novosibirsk Oral Geo Oly 7.5)