G285 2021 MC10B
Problem 1
Find
Problem 2
If , and
, and
, what is
?
Problem 3
A convex hexagon of length is inscribed in a circle of radius
, where
. If
, and
, find the area of the hexagon.
Problem 4
Find the smallest such that:
Problem 5
A principal is pushing out an emergency COVID-19 alert to his school of teachers and
students. Suppose the announcement is first approved by his
aides. Then, each of the aides share the announcement to
teachers and
students, where
and for every aide
. Moreover,
, where
is the round number ( for the aides releasing info it is round 1, then round 2....) After every round
, some
teachers in the previous round share the announcement to a new group of
teachers and
students, where
. How many rounds will it take until the entire school is informed? Assume that after all teachers are informed,
, but
still grows as if
.
Problem 6
Let a plane parallel to the horizontal slice a sphere with radius at a random location to create a cross section, and two partial spheres. For what minimum radius
will the area of the cross-section never be able to exceed the sum of the outer surface areas of the partial spheres?