14.1.2. Tutorial 2: Part II

Try these first

1. 1.

   What is the relationship between arithmetic mean and geometric mean for two numbers?

   Hint

   Think about how averages work.

2. 2.

   Can you represent the number 6 as a sum of distinct powers of two?

   Hint

   Consider the binary equivalent.

3. 3.

   The arithmetic mean of two positive numbers is always:

   • Less than their geometric mean
   • Equal to their geometric mean
   • Greater than or equal to their geometric mean

   Hint

   Think about the two numbers in question.

4. 4.

   True or False: Every integer has a distinct binary representation.

   • True
   • False

   Hint

   Consider how binary digits work.

5. 5.

   Using your understanding of the AM-GM inequality, prove that for any three positive numbers, the arithmetic mean will always be greater than the geometric mean.

   Hint

   Start by proving for two, then extend your logic.

6. 6.

   Create a logical argument or a proof for how many guests can be at a party where everyone is not a celebrity.

   Hint

   Think about the implications of 'knowing' someone.