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$(4+1)(8+1)(2+1)(2+1) = 5 \times 9 \times 3 \times 3 = 45 \times 9 = 450 - 45 = \boxed{405}$

Let's see the equation of the line first. Choose two arbitrary points, say $(0,1), (1,0).$ We find the slope to be $1.$ The y-intercept is also $1,$ so our equation is $y = x + 1$(if it wasn't already trivial.)

We see that it cannot be less, and the line is straight not dotted, so the equation is $y \geq x + 1.$

$x^2 - 7x - 44 = 0$

$(x-11)(x+4) = 0$

$x = 11; x = -4.$

$3(-4) + 2(7) - 5(-1) + 8(3) - 4(-5) = -12 + 14 + 5 + 24 + 20 = 51$

First thing that comes to mind is a marathon, but marathons are held inside central america. Iirc, the answer is decalthon.

6x^2  - 31x  +  42   =  0

Quadratic Formula

x   =  ( 31  -  sqrt  ( 31^2  -4 * 6 * 42) ) / (2* 6)   =   ( 31  -  sqrt  (-47) )  /  12  =

31/12   -  [ sqrt (47)/ 12] i

Also

31/12  +   [ sqrt (47) / 12 ] i

f(3)   =  f(2)  + f(1)   =  13

f (4)  =  f(3)  + f(2)  =  23

f ( 5)  =   f(4) + f(3)    =  36

f ( 6)  =  f(5)  + f(4)   =  59

x^2 - 18x - 33 = 0

(x-9)^2 - 114 = 0

(x-9)^2 = 114

(x-9) = sqrt(114)

*Note that normally we have to account for negatives, but -sqrt(114) will make x a negative number which is not possible. 

x = sqrt(114) + 9

=^._.^=

sqrt(1000) = 31.6227766017

So the number of squares from 1 to 1000 is 31. (1^2, 2^2, ..., 31^2)

Out of that list, 16 of them are odd squares. 

500 numbers from 1 to 1000 are odd. 

So 500 - 16 are odd numbers that aren't squares. 

=^._.^=

Edit:

Not square, but square free. 

Not sure how to do this question. 

I hope someone else can help more. 

Maybe casework, but 1000 is quite large.

=^._.^=

(24x^4  + 6x^3  + 4x^2  + 7x  - 5)     ( 6x^3  + 3x^2  - 3x   + 4)

The  x^3  terms will  come from the sum of  these  multiplications

(6x^3) (4)   +    (4x^2) (-3x)    +  (7x)(3x^2)  +  (-5)(6x^3)  =

[ 24    - 12  +  21  - 30  ]  x^3   =

        3x^3

f +  g =    x - 2    ......so

g   =   x - 2   -  f

g =  x  - 2   - ( x^3 + 5x  -  8)

g  =   x - 2  - x^3  -  5x  +  8

g =  -x^3  -  4x  +  6

Using the  change of base rule

Let  x =  log b/ log a        y  = log c / log b        z =   log  a / log c 

x +  y  +  z  =  0

x + y    =   -z               square both sides

x^2 + y^2  + 2xy   =  z^2

x^2 + y^2   =  z^2   - 2xy

x^3  +  y^3   +  z^3   =

(x + y)  ( x^2  - xy + y^2)  +  z^3   =  

(-z) ( z^2 - 2xy - xy)  + z^3    = 

(-z) ( z^2  -  3xy)  + z^3   = 

-z^3  + 3xyz  + z^3    =

3xyz   =

3  [ log b / log a  ]  [ log c / log b   ]   [ log a / log c]  =

3 [ log a/ loga ]   [ log b  /log b  ] [ log c / log c  ]   =

3 [ 1] [ 1 ] [ 1 ] = 

3

If f(0)  =  0.....then  c  =  0

And  we  have  this system

a ( 1)^2  + b(1)   = 2              ⇒   a  +   b  =  2    ⇒    -2a - 2b  =   -4      (1)

a(2)^2  +  b(2)  = 9               ⇒  4a  +2b   =  9     (2)

Add (1)  and (2)

2a  =  5

a   = 5/2

b = 2 - a   =   2 - (5/2)   =  -1/2

f(3)   =  (5/2)(3)^2  - (1/2)(3)   =    

45/2 -  3/2  =

42/2   =

21

4 ( x + 7) ( 12- x)

The roots  are    x = -7     and  x  =12

Because of  symmetry.....the  max  occurs at  x = ( 12  +  -  7)  / 2  =   5/2 =   2.5

Subbing  this back into the original expression , the  max is

4 ( 2.5 + 7) ( 12 - 2.5)   =

4 ( 9.5) ( 9.5)   =

361 

~ never mind ~

The possible  sums are  all  multiples of  3

(18, 21, 24 ......78, 81)   =

(81 - 18)  / 3    +  1   =

63/3  +  1  =

21  +  1  =

22 distinct sums

2x^2 - 3x + 29     divide through by 2

x^2 - 3/2x + 29/2       roots   r and s   SUM to -3/2   and multiply  to  29/2

Now:

(r+s)^2   = r^2 + s^2 + 2rs    sub in the values above

   (-3/2)^2 = r^2 + s^2  + 2 ( 29/2) 

r^2 + s^2 =   (-3/2)^2 - 2 ( 29/2)            You can finish !            

There are 10 elements, so the answer is C(10,3) = 120.

Multiplying a polynomial by a non-zero constant does  not  change its  degree

So  

If  f ( x)  is degree 4     and  h(x)  is degree 6 ,  then  g(x)   must  be   degree 6

think thats correct 

The plank is in static equilibrium ....the torque at B must be balanced

From B perspective

  F * d  =    F_d  3            F_{d  =}  force of diver = ma

2300  = 3 * ma

              = 3 * m (9.81)            m =  78.15 kg

editing 

We can use casework to solve this question.

1*1 rectangles: 12

1*2 rectangles: 9

1*3 rectangles: 6

1*4 rectangles: 3

2*1 rectangles: 8

2*2 rectangles: 6

2*3 rectangles: 4

2*4 rectangles: 2

3*1 rectangles: 4

3*2 rectangles: 3

3*3 rectangles: 2

3*4 rectangles: 1

So, the answer to this question is 12+9+6+3+8+6+4+2+4+3+2+1=60.