MathDB

p21. If

f = \cos(\sin (x))

. Calculate the sum

\sum^{2021}_{n=0} f'' (n \pi)

.
p22. Find all real values of

A

that minimize the difference between the local maximum and local minimum of

f(x) = \left(3x^2 - 4\right)\left(x - A + \frac{1}{A}\right)

.
p23. Bessie is playing a game. She labels a square with vertices labeled

A, B, C, D

in clockwise order. There are

7

possible moves: she can rotate her square

90

degrees about the center,

180

degrees about the center,

270

degrees about the center; or she can flip across diagonal

AC

, flip across diagonal

BD

, flip the square horizontally (flip the square so that vertices A and B are switched and vertices

C

and

D

are switched), or flip the square vertically (vertices

B

and

C

are switched, vertices

A

and

D

are switched). In how many ways can Bessie arrive back at the original square for the first time in

3

moves?
p24. A positive integer is called happy if the sum of its digits equals the two-digit integer formed by its two leftmost digits. Find the number of

5

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

Problem Statement