7

Part of 2019 Novosibirsk Oral Olympiad in Geometry

Problems(3)

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

Source:

Cut a square into eight acute-angled triangles.

geometryacutesquare

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

Source:

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

geometryacutesquare

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

Source:

X,Y

two convex polygons, such that

X

is contained inside

Y

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)}.

geometrygeometric inequalityconvex polygon