MathDB

p29. Consider pentagon

ABCDE

. How many paths are there from vertex

A

to vertex

E

where no edge is repeated and does not go through

E

.
p30. Let

a_1, a_2, ...

be a sequence of positive real numbers such that

\sum^{\infty}_{n=1} a_n = 4

. Compute the maximum possible value of

\sum^{\infty}_{n=1}\frac{\sqrt{a_n}}{2^n}

(assume this always converges).
p31. Define function

f(x) = x^4 + 4

. Let

P =\prod^{2021}_{k=1} \frac{f(4k - 1)}{f(4k - 3)}.

Find the remainder when

P

is divided by

1000

.
p32. Reduce the following expression to a simplified rational:

\cos^7 \frac{\pi}{9} + \cos^7 \frac{5\pi}{9}+ \cos^7 \frac{7\pi}{9}

PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

R8