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And it shows that everyone is struggling on it ...

Everyone has to answer that question ....

x = initial amount of people at the dance and

      Initially    .40 x = freshmen_o 

   then add 15 freshmen

.50 (x+15) = freshmen_o + 15

 .50(x+15 ) = .40x + 15

  .5x + 7.5 = .40x + 15

   .1x = 7.5

      x = 75    people at the dance initially .....        after 15 more freshmen arrive there is 90 people at the dance

What about (b)?

The remainder when $s_{100}$ is divided by 5 is 4.

The probability is 1/8.

The probability is 2/5.

$p_1(2) + p_2(2) + \dots + p_k(2) = 18$

Multiplying a polynomial by a constant does not change its degree.....so   f + g will be degree 5

b = 1/4 g

1/3 b   + 1/2 g = 84

1/3 (1/4 g )   + 1/2 g = 84

7/12 g = 84

g = 144     then   b  = 36          b + g = _______________

1 x³     + 8x³   =   9 x³

.6x = 22

x = 36 2/3

.55 ( 36 2/3) =121/6

Hmmmmmm.....   is this perhaps 64 feet or something else ?

Since tigers and giraffes have the same number of feet and heads     g = t

62 feet =   2k  + 4 ( 2g )         

22 heads =  k + g + g = 22 = k + 2g    <===== multiply by -2 and add to top equation to get

18 = 4 g

Integers     x-1     x       x+1

(x+1)(x-1) =3x + 17

x^2 -1 -17-3x = 0

x^2 -3x - 18 = 0

(x-6)(x+3)                  x = 6     or -3            5 6 7           ( or    -4 -3 -2   <==== negative integers solution)

No such  'c' exists

-5   +     +5    = 0

The probability is 1/4 .

\(f(x) = \frac{x - \sqrt{3}}{x\sqrt{3} + 1} \\~\\ f(f(x)) = \frac{\frac{x - \sqrt{3}}{x\sqrt{3} + 1} - \sqrt{3}}{\frac{x - \sqrt{3}}{x\sqrt{3} + 1}\sqrt{3} + 1}\\ \text{This simplifies to}\\ f(f(x)) =\frac{x+\sqrt3}{1-\sqrt3x}\\~\\ f(f(f(x)) )=\frac{\frac{x+\sqrt3}{1-\sqrt3x}+\sqrt3}{1-\sqrt3\frac{x+\sqrt3}{1-\sqrt3x}}\\~\\ \text{This simplifies to}\\ f(f(f(x)) )=x\\~\\ \text{now there is a loop happening.}\\~\\ f^0(x)=x\\ f^1(x) = \frac{x - \sqrt{3}}{x\sqrt{3} + 1} \\~\\ f^2(x)=\frac{x+\sqrt3}{1-\sqrt3x}\\~\\ f^3(x)=x\\~\\ \)

2012=2(mod3)

so it  seems that

\(f^{2012}(x)=\frac{x+\sqrt3}{1-\sqrt3x}\\~\\\)

Coding

f(x) = \frac{x - \sqrt{3}}{x\sqrt{3} + 1} \\~\\

f(f(x)) = \frac{\frac{x - \sqrt{3}}{x\sqrt{3} + 1} - \sqrt{3}}{\frac{x - \sqrt{3}}{x\sqrt{3} + 1}\sqrt{3} + 1}\\
\text{This simplifies to}\\
f(f(x)) =\frac{x+\sqrt3}{1-\sqrt3x}\\~\\

f(f(f(x)) )=\frac{\frac{x+\sqrt3}{1-\sqrt3x}+\sqrt3}{1-\sqrt3\frac{x+\sqrt3}{1-\sqrt3x}}\\~\\
\text{This simplifies to}\\
f(f(f(x)) )=x\\~\\
\text{now there is a loop happening.}\\~\\
f^0(x)=x\\
f^1(x) = \frac{x - \sqrt{3}}{x\sqrt{3} + 1} \\~\\

f^2(x)=\frac{x+\sqrt3}{1-\sqrt3x}\\~\\

f^3(x)=x\\~\\

The prime factors of 66 are 2.3.11, so 4a5b72 must be divisible by these factors.

 Clearly it is divisible by 2.

For it to be divisible by 3 we must have a+b as a multiple of 3.

Try the simplest:  a = 0, b = 3:   405372/66 = 6142.

Assuming the sum of a and b is unique, then a+b = 3

To decrease a number by 40% multiply it by 0.6 (60% = 100% - 40%).

To decrease a number by 20% multiply it by 0.8 (80% = 100% - 20%)

So result = 0.8*(0.6*75) = ...  you can crunch the numbers.

The domain is [-2,inf).

Is there any other way to solve it?

The surface area of the hemisphere is 8/3*pi.