CPhill

a11 - a10 = a9

1 - 4 = -3 

a10 - a9  = a8

4  - - 3  = 7

a9 - a8 = a7

-3 - 7  = -10

a8 - a7 = a6

7 - -10 = 17

Simplify as

x^2 -mx + 14 =  0

Note that  the possible factorizations are

( x - 7) (x - 2)      m =  9

( x + 7) (x + 2)    m = -9

(x - 14) ( x -1)    m = 14

(x + 14) ( x + 1)  m = -14

a^3 - 1/a^3  =   (a -1/a) (a^2 + 1 + 1/a^2) =  (0) ( a^2 + 1 + 1/b^2)  =  0

Let the roots be a,b

Product of the roots  = ab =  -4   →  2ab = -8

Sum of the roots =  a + b = -12     square both sides

a^2 + b^2 + 2ab = 144

a^2 + b^2 - 8 =144

a^2 + b^2 =  152

A

                X

                     5

B                        C

            15

Since BXA = 90  then  BXC also = 90

So BXC is aright triangle such that

BC^2 - CX^2  = BX^2

15^2 - 5^2  = BX^2

200 = BX^2

BX  = sqrt 200  =   10sqrt 2  ≈  14.14

a + b = 4      square both sides

a^2 + 2ab + b^2  =16

a^2 + b^2 + 2ab = 16

6   + 2ab =  16

2ab =  10

ab = 5

a^3 + b^3  = (a + b) (a^2 + b^2  - ab) =  (4) (6 - 5)  =   4

                      A

               30       30

B                      D                        C

Impossible

If AD is a bisector, then CAD  =  BAC / 2 =   30

So CAD cannot = 45

Since

    (1)   =  (1)       set the equations  = 

x^2 + (y-1)^2  = (x-1)^2 + y^2     simplify

x^2 + y^2 - 2y + 1 =  x^2 -2x + 1 + y^2

-2y = -2x

x = y

So

x^2 + ( x - 1)^2  =  1

x^2 + x^2 -2x + 1   = 1

2x^2 - 2x  =  0

x^2 - x = 0

x ( x -1) = 0

x = 0       x   = 1

y = 0      y = 1

The intersection points are   ( 0, 0)  and (1, 1)

The distance  between them =   sqrt [1^2 + 1^2]  = sqrt 2

Let B = (0,0)

Let C = (32,0)

Let A  be the apex

The line through AB has the equation

y = (3/5)x

The line through CA has the equation

y = (-1/4) (x - 32)  =  (-1/4)x + (-1/4)

To find the x coordiante of A, set the lines equal

(3/5)x = (-1/4)x + 8

(3/5 + 1/4)x  = 8

(17/20) x = 8

x = 160 / 17

y = (3/5) (160/17)  =  480/85   = 96/17 =  the height of the triangle

[ABC ]  =  (1/2) (32) (96/17)  = 1536/17 ≈  90.35

xz = x + z                    yz = 2 ( y + z)

xz - x = z                     yz = 2y + 2z

x ( z  -1) = z                yz - 2y  = 2z

x = z / ( z -1)                  y ( z - 2)   =2z

                                    y = (2z) / (z -2)

xy = x + y

z/ (z-1)* (2z) ( z -2)  =   z/(z -1) + (2z) (z -2)

[2z^2] / [ (z-1)(z -2)] =   [  z (z -2) +  2z ( z -1)] / [ (z -1) ( z -2) ]

2z^2  = z^2 - 2z + 2z^2 - 2z

2z^2  = 3z^2 - 4z

z^2 - 4z  =  0

z ( z - 4)  = 0 

Reject z  = 0

z =  4