2

Part of 2022 Novosibirsk Oral Olympiad in Geometry

Problems(3)

perimeter of quad (2022 Novosibirsk Oral Geo Oly 7.2 8.1)

Source:

A quadrilateral is given, in which the lengths of some two sides are equal to

1

4

. Also, the diagonal of length

2

divides it into two isosceles triangles. Find the perimeter of this quadrilateral.

geometryperimeter

reflections on billiard (2022 Novosibirsk Oral Geo Oly 8.2)

Source:

A ball was launched on a rectangular billiard table at an angle of

45^o

to one of the sides. Reflected from all sides (the angle of incidence is equal to the angle of reflection), he returned to his original position . It is known that one of the sides of the table has a length of one meter. Find the length of the second side. https://cdn.artofproblemsolving.com/attachments/3/d/e0310ea910c7e3272396cd034421d1f3e88228.png

geometryreflectionrectangle

min total length by 4 integer line segments (2022 Novosibirsk Oral Geo Oly 9.2)

Source:

Faith has four different integer length segments. It turned out that any three of them can form a triangle. What is the smallest total length of this set of segments?

geometrycombinatorial geometrytriangle inequality