1. Grogg, Lizzie, Alex, and Winnie each own a hat marked with the first letter of their name. How many ways can the four of them each wear one of these four hats so no one is wearing a hat marked with with the first letter of their
name?
2. A rectangular prism has a square base with side length s units. It also has the curious property that the numerical
value of the sum of its edge lengths (in units) is equal to the numerical value of its surface area (in square units).
If one edge of the prism has length 8 units and s 6= 8, then s can be written in the form a +
√
b, where a and b are

integers. Find a + b.
3. Find the smallest integer n such that the product of the divisors of n is equal to n
9
.
4. Five baby orangutoads are weighed in pairs. The recorded weights (in grams) are as follows:

165, 173, 176, 177, 180, 182, 185, 188, 193, 197.

Which of these ten weights represents the combined weight of the second-lightest and second-heaviest orangutoads,
in grams?
5. A positive integer is called an up-down integer if any two consecutive digits of the integer differ by exactly 1. How
many nine-digit up-down integers are there with first and last digits both equal to 5?
6. R&G’s farm has a pen shaped like a regular nonagon (a 9-sided polygon), where each side has length 1 meter. A
narwolf is on a leash of length 3 meters, the other end of which is tied to a vertex of this pen. If the narwolf is not
allowed to go inside the pen and there are no other obstacles in the way, the area of the region where the narwolf
can roam while staying leashed to the pen (in square meters) can be written as m
n
π, where m and n are relatively

prime positive integers. Find m + n.
7. The exact value of the expression
s
102 +
r
104 +
q
108 +
p
1016 + · · ·

can be written in the form a +
√
b, where a and b are integers. Find a + b.
8. Find the number of positive integers n, 1 ≤ n ≤ 10000, such that n is divisible by b
√
nc.
Note: For a real number x, bxc is defined as the greatest integer that is less than or equal to x.