MathDB

p7 Cindy cuts regular hexagon

ABCDEF

out of a sheet of paper. She folds

B

over

AC

, resulting in a pentagon. Then, she folds

A

over

CF

, resulting in a quadrilateral. The area of

ABCDEF

k

times the area of the resulting folded shape. Find

k

.
p8 Call a sequence

\{a_n\} = a_1, a_2, a_3, . . .

of positive integers Fib-o’nacci if it satisfies

a_n = a_{n-1}+a_{n-2}

for all

n \ge 3

. Suppose that

m

is the largest even positive integer such that exactly one Fib-o’nacci sequence satisfies

a_5 = m

, and suppose that

n

is the largest odd positive integer such that exactly one Fib-o’nacci sequence satisfies

a_5 = n

. Find

mn

.
p9 Compute the number of ways there are to pick three non-empty subsets

A

B

, and

C

\{1, 2, 3, 4, 5, 6\}

, such that

|A| = |B| = |C|

and the following property holds:

A \cap B \cap C = A \cap B = B \cap C = C \cap A.

R3