By: Xavier Blackwell-Lipkind

Note: The problems below, listed in no particular order, represent a wide range of difficulties. All are original. Exactly one problem is unsolved.

 

1. Solve the following equation for x.

x=sqrt(1+sqrt(1+sqrt(1+...)))

 

2. What do the following numbers have in common? (Hint: they can be expressed as a non-recursive sequence.)

0, 4, 18, 48, 100, 180,...

 

3. What is the last (ones) digit of (75)^(∞)?

 

4. There is a 20% chance that a randomly selected Conard student lives in a blue house. Find the probability that of five Conard students polled in succession, at least one lives in a non-blue house.

 

5. Simplify: (e^(i𝜋 + 1)) / (-𝜋).

6. How many prime triplets exist, such that n, n+2, and n+6 or n, n+4, and n+6 are all prime? How about prime doublets (n, n+2)?