CPhill

Colin and Sarah must have  1 - 1/9   =   8/9  of the money

And Colin  has  3 / [ 3 + 1 ]  =  3/4   of this

So.....the proportion that Colin gets  is

8/9  *  3/4   =  

24/36   =

2/3

Thanks, Melody    !!!!

Here are the two matrices that we need :

    P                     S

3  4  1  1            8    8               3*8  + 4*10 + 1*7 + 1*12        3*8 + 4*9 + 1*8  + 1*13

6  6  4  6           10   9     =        6*8  + 6*10 + 4*7 + 6*12        6*8 + 6*9  + 4*8  + 6*13   =

2  2   5 4            7    8               2*8  + 2*10 + 5*7 + 4*12        2*8 + 2*9 + 5*8  + 4*13

                          12 13

83     81

208   212

119   126 

Well, CU......your hint was a GREAT one   !!!

I wouldn't have seen how to proceed without it  !!!

Using CU's hint :

(p + q+ r)  + (qx + ry)  + (pz + rx)  + ( py + qz)  =  x + y + z

-3   +  x (q + r)  + y (p + r) + z(p + q)  =  x + y + z

-3   + x(-3 - p)  + y (-3 - q)  + z (-3 - r)  = x + y + z

-3  - 3x  - px  - 3y - qy  - 3z - rz  =  x + y + z

-3 - 3 ( x + y + z)  -  ( px + qy + rz)  = x + y + z

-3  - 3(x + y + z)  -  (1)  = x + y + z

-3 - 1  =  4 (x + y + z)

-4  =  4 (x + y + z)

-1  = x + y + z

Thanks for that hint, CU......that's pretty crafty   !!!

If   the vertex  is  (3,2)    and  it contains the point   (1,0).....then x  = 1 is a  root

So...by symmetry......(5, 0) is also a root

So....we can find  "a"  thusly

0  =  a(1 - 3)^2  + 2

-2  =  a(-2)^2

-2  = 4a

a  =  -2/4  =  -1/2

And by Vieta......

The sum of the roots   -b/a

So

1 + 5  = - b / ( -1/2)

6 = 2b

b = 3

And the product of the roots  = c/a

So

5*1  =  c /(-1/2)

5 = -2c

c = -5/2

So

(a,b , c)   =  ( -1/2, 3, -5/2)

Here's the graph :  https://www.desmos.com/calculator/yd99xbrpsg

We can solve this  for  "a"

-3  = a (-4 -2)^2  + 1

-3 = a ( -6)^2 + 1

-4  = 36a

a  = -4/36  = - 1/9

So....we have this form

y  = (-1/9( ( x - 2)^2  + 1       to find the zeroes....let y  = 0

0  = (-1/9)(x - 2)^2 + 1

-1  = (-1/9)(x - 2)^2       multiply through by -9

9 =  ( x - 2)^2        take both roots

±√9  = x - 2

±3  =  x - 2    add 2 to both sides

±3 + 2  =  x

So....the roots are

x =  3 + 2  = 5  = m         and   x  =  -3 + 2  =  -1  = n

So

m - n  =   5 - (-1)   =  6

LOL!!!!!......

x + 1/x  = 7        square both sides

(x + 1x) (x + 1/x)  = 49

x^2  +  x (1/x)  + x (1/x)  +  1/x^2  = 49     

x^2  +  1  + 1  + 1/x^2  =  49     subtract  1 from both sides and rearrange the left side

x^2 + 1/x^2  + 1  =   48    Voila  !!!!