3

Part of 2019 Novosibirsk Oral Olympiad in Geometry

Problems(3)

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

Source:

Equal line segments are marked in triangle

ABC

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

geometryanglesequal segments

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

Source:

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

ratiopaper foldingfoldinggeometrysquare

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

Source:

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

geometryareasquare