CPhill

x^2 - x - 2 =  (x - 2) ( x + 1)

Multiply through by the factorization and we get

2x + 11 =  A(x + 1)  + B( x  -2)

2x + 11 = (A + B)x + (A - 2B)

Equating terms

A + B  = 2     →   B = 2 - A    (1)

A - 2B  = 11     (2)

Sub (1)  into (2)

A - 2 ( 2 -A)  = 11

A - 4 + 2A  = 11

3A = 15

A = 5

B = 2 - 5  =  -3

Law of Cosines

BC^2  = BA^2  + AC^2  -2 ( BA * AC) cos (DAE)

18^2 = 10^2 + 14^2  - 2 ( 10 * 14)  cos (DAE)

[ 18^2 - 10^2 - 14^2 ] / [ - 2 (10 * 14) ] =  cos (DAE)

-1/10  = cos (DAE)

Again

DE^2  =  AD^2 + AE^2 - 2(AD * AE) cos (DAE)

DE^2  = 28^2 + 20^2 - 2(28 * 20) (-1/10)

DE^2  = 1296

DE  =  sqrt [1296]    = 36

                               A

                          1

                     F

                 2

           D                                     E

B                                                             C

FE patallel to DC so triangles AFE  and ADC  are similar

AF / FE  = AD / DC

1 / FE  = 3 / DC

FE = (1/3)DC

3FE = DC

FE parallel to DC   and DE parallel to BC  so triangle FDE  is similar to triangle DBC

FD / FE = BD /  DC

FD / FE = BD /  (3FE)

FD = BD /  3

3FD = BD

3*2 = BD =  6 

1. Call the center of the circle O

[AOP ] = (1/2) (2)(4)  =  4

OP =  sqrt [ 4^2 + 2^2] = sqrt [ 20 ] = 2sqrt 5

Height of triangle AOP can  be found as

4 = (1/2) 2 sqrt 5  *  h

h =  4/sqrt 5  =   AB /  2

AB  =   2 * 4/sqrt 5  =  8 / sqrt 5

3.

Angle XTA = 180 - 47 =  133

Angle XUA = 180  - 65  = 115

Minor arc TA =  2*47 =  94

Minor arc UA = 2*65 = 130

Arc TU =  360 - 94 - 130 =  136

Angle TAU =  136 / 2 =  68

Angle QXR  =  360  - 133  - 115  -  68    = 44

Side of the square  = [ product of the legs ] / [ sum of the legs ]  =  

[ 6 * 6 ]  / [ 6 + 6 ] = 

36 / 12   = 

3

Friend 1        Friend 2

11                    1

10                    2

.

.

.

1                   11

11  ways

f(1) =  -1

f(2) =   0

All   the other terms evaluate to 0   because the inside function will approach (but never reach) y = 1 since this is the horizontal  asymptote of that  function

So.....the sum  =  -1

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