MathDB

3

collinearities by 2 squares (2023 Novosibirsk Oral Geo Oly 7.7)

ABCD

and

BEFG

are located as shown in the figure. It turned out that points

A, G

and

E

lie on the same straight line. Prove that then the points

D, F

and

E

also lie on the same line. https://cdn.artofproblemsolving.com/attachments/4/2/9faf29a399d3a622c84f5d4a3cfcf5e99539c0.png

2 // lines intersect a square (2023 Novosibirsk Oral Geo Oly 8.7)

A square with side

1

is intersected by two parallel lines as shown in the figure. Find the sum of the perimeters of the shaded triangles if the distance between the lines is also

1

. https://cdn.artofproblemsolving.com/attachments/9/e/4e70610b80871325a72e923a0909eff06aebfa.png

<EDC = 2< BCD = 2< CAB, 100-60-20 (2023 Novosibirsk Oral Geo Oly 9.7)

Triangle

ABC

is given with angles

\angle ABC = 60^o

and

\angle BCA = 100^o

. On the sides AB and AC, the points

D

and

E

are chosen, respectively, in such a way that

\angle EDC = 2\angle BCD = 2\angle CAB

. Find the angle

\angle BED

.

6

3

<ADB+<BEC+<CFA=? BA=BC, AD=AC, BE=BA , CF=CB (2023 Novosibirsk Oral Geo Oly 7.6)

An isosceles triangle

ABC

with base

AC

is given. On the rays

CA

,

AB

and

BC

, the points

D, E

and

F

were marked, respectively, in such a way that

AD = AC

,

BE = BA

and

CF = CB

. Find the sum of the angles

\angle ADB

,

\angle BEC

and

\angle CFA

.

area bisector of mushroom-gon (2023 Novosibirsk Oral Geo Oly 8.6)

Let's call a convex figure, the boundary of which consists of two segments and an arc of a circle, a mushroom-gon (see fig.). An arbitrary mushroom-gon is given. Use a compass and straightedge to draw a straight line dividing its area in half. https://cdn.artofproblemsolving.com/attachments/d/e/e541a83a7bb31ba14b3637f82e6a6d1ea51e22.png

angle wanted, 2 quarter-circles (2023 Novosibirsk Oral Geo Oly 9.6)

Two quarter-circles touch as shown. Find the angle

x

. https://cdn.artofproblemsolving.com/attachments/b/4/e70d5d69e46d6d40368f143cb83cf10b7d6d98.png

5

2

quad inside quad, compare diagonals' sum (2023 Novosibirsk Oral Geo Oly 7.5)

One convex quadrilateral is inside another. Can it turn out that the sum of the lengths of the diagonals of the outer quadrilateral is less than the sum of the lengths of the diagonals of the inner?

circle inscribed in convex polygon (2023 Novosibirsk Oral Geo Oly 9.5)

A circle of length

10

is inscribed in a convex polygon with perimeter

15

. What part of the area of this polygon is occupied by the resulting circle?

4

2

5 congruent triangles inside convex 5gon (2023 Novosibirsk Oral Geo Oly 7.4)

Inside the convex pentagon

ABCDE

, a point

O

was chosen, and it turned out that all five triangles

AOB

,

BOC

,

COD

,

DOE

and

EOA

are congrunet to each other. Prove that these triangles are isosceles or right-angled.

isosceles trapezoid wanted (2023 Novosibirsk Oral Geo Oly 9.4)

In a trapezoid, the length of one of the diagonals is equal to the sum of the lengths of the bases, and the angle between the diagonals is

60

degrees. Prove that this trapezoid is isosceles.

3

2

rectangle cut into 10 squares (2023 Novosibirsk Oral Geo Oly 7.3)

The rectangle is cut into

10

squares as shown in the figure on the right. Find its sides if the side of the smallest square is

3

.https://cdn.artofproblemsolving.com/attachments/e/5/1fe3a0e41b2d3182338a557d3d44ff5ef9385d.png

line segments from 5 points (2023 Novosibirsk Oral Geo Oly 8.3)

Points

A, B, C, D

and

E

are located on the plane. It is known that

CA = 12

,

AB = 8

,

BC = 4

,

CD = 5

,

DB = 3

,

BE = 6

and

ED = 3

. Find the length of

AE

.

2

1

compare areas, midpoints in square (2023 Novosibirsk Oral Geo Oly 7.2)

In the square, the midpoints of the two sides were marked and the segments shown in the figure on the left were drawn. Which of the shaded quadrilaterals has the largest area? https://cdn.artofproblemsolving.com/attachments/d/f/2be7bcda3fa04943687de9e043bd8baf40c98c.png

1

2

cut a 6x6 square into tiles 1x2 (2023 Novosibirsk Oral Geo Oly 7.1)

Let's call a corner the figure that is obtained by removing one cell from a

2 \times 2

square. Cut the

6 \times 6

square into corners so that no two of them form a

2 \times 3

or

3 \times 2

rectangle together.

CD, BE divide ABC in 4 parts of same area ? (2023 Novosibirsk Oral Geo Oly 9.1)

In the triangle

ABC

on the sides

AB

and

AC

, points

D

and E are chosen, respectively. Can the segments

CD

and

BE

divide

ABC

into four parts of the same area? https://cdn.artofproblemsolving.com/attachments/1/c/3bbadab162b22530f1b254e744ecd068dea65e.png

2023 Novosibirsk Oral Olympiad in Geometry

Subcontests

collinearities by 2 squares (2023 Novosibirsk Oral Geo Oly 7.7)

2 // lines intersect a square (2023 Novosibirsk Oral Geo Oly 8.7)

<EDC = 2< BCD = 2< CAB, 100-60-20 (2023 Novosibirsk Oral Geo Oly 9.7)

<ADB+<BEC+<CFA=? BA=BC, AD=AC, BE=BA , CF=CB (2023 Novosibirsk Oral Geo Oly 7.6)

area bisector of mushroom-gon (2023 Novosibirsk Oral Geo Oly 8.6)

angle wanted, 2 quarter-circles (2023 Novosibirsk Oral Geo Oly 9.6)

quad inside quad, compare diagonals' sum (2023 Novosibirsk Oral Geo Oly 7.5)

circle inscribed in convex polygon (2023 Novosibirsk Oral Geo Oly 9.5)

5 congruent triangles inside convex 5gon (2023 Novosibirsk Oral Geo Oly 7.4)

isosceles trapezoid wanted (2023 Novosibirsk Oral Geo Oly 9.4)

rectangle cut into 10 squares (2023 Novosibirsk Oral Geo Oly 7.3)

line segments from 5 points (2023 Novosibirsk Oral Geo Oly 8.3)

compare areas, midpoints in square (2023 Novosibirsk Oral Geo Oly 7.2)

cut a 6x6 square into tiles 1x2 (2023 Novosibirsk Oral Geo Oly 7.1)

CD, BE divide ABC in 4 parts of same area ? (2023 Novosibirsk Oral Geo Oly 9.1)

2023 Novosibirsk Oral Olympiad in Geometry

Subcontests

collinearities by 2 squares (2023 Novosibirsk Oral Geo Oly 7.7)

2 // lines intersect a square (2023 Novosibirsk Oral Geo Oly 8.7)

&lt;EDC = 2&lt; BCD = 2&lt; CAB, 100-60-20 (2023 Novosibirsk Oral Geo Oly 9.7)

&lt;ADB+&lt;BEC+&lt;CFA=? BA=BC, AD=AC, BE=BA , CF=CB (2023 Novosibirsk Oral Geo Oly 7.6)

area bisector of mushroom-gon (2023 Novosibirsk Oral Geo Oly 8.6)

angle wanted, 2 quarter-circles (2023 Novosibirsk Oral Geo Oly 9.6)

quad inside quad, compare diagonals&#039; sum (2023 Novosibirsk Oral Geo Oly 7.5)

circle inscribed in convex polygon (2023 Novosibirsk Oral Geo Oly 9.5)

5 congruent triangles inside convex 5gon (2023 Novosibirsk Oral Geo Oly 7.4)

isosceles trapezoid wanted (2023 Novosibirsk Oral Geo Oly 9.4)

rectangle cut into 10 squares (2023 Novosibirsk Oral Geo Oly 7.3)

line segments from 5 points (2023 Novosibirsk Oral Geo Oly 8.3)

compare areas, midpoints in square (2023 Novosibirsk Oral Geo Oly 7.2)

cut a 6x6 square into tiles 1x2 (2023 Novosibirsk Oral Geo Oly 7.1)

CD, BE divide ABC in 4 parts of same area ? (2023 Novosibirsk Oral Geo Oly 9.1)

<EDC = 2< BCD = 2< CAB, 100-60-20 (2023 Novosibirsk Oral Geo Oly 9.7)

<ADB+<BEC+<CFA=? BA=BC, AD=AC, BE=BA , CF=CB (2023 Novosibirsk Oral Geo Oly 7.6)

quad inside quad, compare diagonals' sum (2023 Novosibirsk Oral Geo Oly 7.5)