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First from A -> C, there are 6 (paths). Then, from C -> B there are 6 (paths). 6*6 = 36.

The  value is  x =    -11/ ( 2 * 1)  =  -11/2  =   - 5.5

Angle Y  = 60

So....using  the Law of  Sines  twice

6  /sin 60    =   XY /sin 70

XY  =  6sin 70   / sin 60   ≈   6.51

And

6 /sin 60  =  ZY /sin 50

ZY  =  6sin 50  / sin 60  ≈  5.3

So.....the perimeter   ≈   6  + 6.51  + 5.3   ≈  17.81  cm

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

So

f (f (-1) )   =  f (-2)   =   3 (-2)^2   -  5 =     12    -  5    =    7

                         C

            8                  16

A                      D             B

                     17

Using  Heron's Formula.....the  semi-perimeter  = ( 17 + 8 + 16)  / 2   =    41/2   =   20.5

And  the  area  =   sqrt [ ( 20.5  ( 20.5 -17) (20.5 - 16) (20.5 - 8)  ] =

sqrt  [  20.5  *  3.5 * 4.5  * 12.5  ]     =

sqrt  [4035.9375 ]

And the  area =   (1/2) AB  * CD

sqrt  [ 4035.3975 ]  =  (1/2) (17)  CD

sqrt [ 4035.3975]  = 8.5 CD

sqrt  [ 4035.3975 ] / 8.5   =  CD  ≈  7.474

Area of  triangle ACD = (1/2) CD  * AD  =   (1/2) (7.474)  *  sqrt  [ 8^2  - 7.474^2 ]   ≈ 10.6615 units^2

                     A

                8

            F

         4

    D                            E

B                                       C

Because   EF  is parallel to  CD,  then  AF/ DF  =  AE  / EC

So     8 /4  =  AE  /EC =  2

So   AE =  2EC

And  because  DE  is parallel to  BC,  then    BD / DA  =  EC / AE

Then

BD / 12 =  EC / 2EC

BD / 12  =  1/2

BD  = 12 * (1/2)  =  6

We  have  the following  set of  moves  (in any order )

( R, R,  R,  R  , U , U , U , U)

Choose  any 4 of  8 positions to put the U's  in   or any 4 of 8 positions to put the R's  in

The  number  of paths =    C (8,4)   =     70

That means the roots are -3 and 8. Putting it in factored form,

\((x+3)(x-8) \\ x^2 - 5x -24\)

Note that  triangles  ABC  and  AST  are similar  

So

AT  /  AC    =   AS / AB

11 / 35   =   AS   / ( AT + BT)

11/  35   =  AS /  ( 11  + 10)

11 /  35 =  AS  /  21          cross-multiply

21 * 11  =  35 * AS

21 * 11   /  35   =   AS   =  33/5   =   6.6

Because of symmetry   about  the  vertex  ....the  other  x intercept =  

(12 -  (18 - 12) , 0  )     =   ( 12  - 6  ,  0  )   =   (6 , 0)        

If  1 -sqrt 2 is  a root  then  so  is 1 + sqrt 2

So...partially,  we  have

(x  - (1  -sqrt 2)  )  ( x  - (1 + sqrt 2) )    =

x^2 - (1 -sqrt 2)x - (1 + sqrt 2)x   +  1  - 2)   =

x^2 - 2x  - 1

Likewise  if     3   = sqrt 5 is a root  then so is 3  -sqrt 5

So

(x - ( 3 + sqrt 5))  ( x - (3  -sqrt 5))  =

x^2 -(3 +sqrt 5) x  - (3  -sqrt 5)) x  + 9 - 5  =

x^2  -6x  +  4

The polynomial is

(x^2   - 2x  -1) ( x^2  - 6x  + 4)  =

x^4  -6x^3  + 4x^2

      -2x^3   + 12x^2 - 8x

                  -    x^2  + 6x  - 4       =

x^4  - 8x^3  + 15x^2  - 2x  -  4   

THANK
THATS ONE ELEGANT SOLUTION!

I think I like EP's  simple solution the  best.....no fuss.....no muss  !!!!!

m/n   *    n/p   * p/q      =   15  * 1/3 * 1/10                   See how things 'cancel out' and you are left with  m/q  ? 

m/q = 1/2

3mn - 6m + 15n + a

mn - 2m + 5n + a/3

(m+5)(n-2) = mn - 2m + 5n - 10

mn - 2m + 5n + a/3 = mn - 2m + 5n - 10

a = -30

=^._.^=

BIG BRAINS 

Yours works OK , too, catmg   !!!!

YOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

THIS IS BIG BRAIN TIME! ANOTHER WAY OF GETTING> THE SAME ANSWER

Another way you could do this is to assume that q = 1. 

If q = 1, and p/q = 1/10, p = 1/10. 

If p = 1/10, and p/n = 3, n = 1/30. 

If n = 1/30, and m/n = 15, m = 1/2. 

CPhill's way is very smart by using substitution. :))

=^._.^=

YOO NOW i get it! Thank you for the edit CPHILL :)

Slight mistake.....check my edit  !!!!!

THANK YOU!!!!!!!'

I am stuck on this step: why will p/q = 1/10 imply  q = (1/10) p?

i'll try to figure it out now. p/q = 1/10. so 10q = p. so q = p/10 which is (1/10) p FIGURED IT OUT

Wait... it's p/q =1/10. so it is 10p = q???? so then p = (1/10) q

m/n  = 15

p/n =    3

This implies  that    m =  5p

And

p/q   = 1/10     impiies  that  q  =  10 p

So

m / q    =    5p  / [ 10 p ]    =   1/2

EDIT .....

OOPS! I misread your problem and thought the 5 was a 2.

For this problem with 1500, it is still similar. 55562 and 35554 are there, so it is 5! / 3! and 5! / 3! = 20 and 20. 20 + 20 = 40

https://web2.0calc.com/questions/in-triangle-and-let-be-the-incenter-the-incircle

(again)