# 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?