CPhill

I'm not that good at these counting problems so I'm not sure of  my error

Slope between the frist two points

[16-7]/[13-1]= 9/12  = 3/4

Equation of line through these two points

y = (3/4) (x -1) + 7           put into  standard form

4y = 3(x -1) + 28

4y = 3x - 3 + 28

3x - 4y + 25  = 0

Using  the partial  formula for the distance  between a  point and  a line,  we want to minimize this

l 3(7) - 4(k) + 25 l  = 0 

l-4k + 46 l  = 0

-4k   = -46

k = 46/4  = 11.5

The integer  values  above and  below this that are closest to 11.5  are  k = 11  and  k =12

Their sum = 23

Here's a graph

Here's what I get.....

4 and  10  =  16  possible pairs

4  and  Jacks  = 16 possible pairs

4  and   Queens = 16 possible pairs

4  and   Kings =  16 possible pairs

5 and 9   =  16 possible pairs

6 and 8  = 16 possible pairs

7s = 4C2 =  6 possible pairs

16*6 + 6 =  102 possible pairs

OOPS!!!!....see my corrected answer

Note

1/ [sqrt a2 + sqrt a1 ]    if we multiply  top/bottom  by the conjugate we have

[ sqrt  a2 - sqrt a1 ] / [ a2 -a1 ] =   [ sqrt a2 - sqrt 1 ] / d

Also

1/ [sqrt a3 + sqrt a2 ]    if we multiply  top/bottom  by the conjugate we have

[ sqrt  a3 - sqrt a2 ] / [ a3 -a2 ] =   [ sqrt a3 - sqrt a2 ] / d

So we   have    [sqrt a2 - sqrt a1 ] / d  +  [sqrt a3 - sqrt a2 ]  /d + [ sqrt a4 -sqrt a3]/ d   +.....+

[sqrt a99 - sqrt a98 ] /d + [sqrt a100 - sqrt a99] / d

So....eventually we just end up with

[  -sqrt a1  + sqrt a100 ]  /  d  =    [ -1 + 25 ] / d

We can find d by noting that

a1  +  99d  =  a100

1 + 99d  = 625

624  =99d

624/99  = d =208/ 33

So...we have

[ 24 ] / [ 208/33]  = 24 * 33 / 208 =  99 / 26

\(1/n(n+3)=blank/n+blank/n+3. \)

1/ [n (n + 3)] = A / n  + B /(n + 3)

Multiply through by  n (n + 3)

1 =  A ( n + 3)  + B(n)

1 = (A + B)n  + 3A

Equate terms

A + B  = 0

3A   =1  →  A =1/3

So B = -1/3

Check     

(1/3) / n   - (1/3) / ( n + 3)  =

[(1/3)(n + 3)  - (1/3) n] / (n (n + 3))

[ (1/3 n  + 1  - 1/3 n ]  / [n ( n + 3)]  = 

1 / [ n ( n + 3) ]

So

first blank = 1/3

second blank  =  -1/3

Find what ????

Let P = (x,y)

PA^2  + PB^2 + PC^2   =

(x - 4)^2  + ( y + 4)^2 + ( x - 3)^2 + ( y -5)^2  + (x + 1)^2 + (y-2)^2

Simplifying this we  get

3 x^2 - 12 x + 3 y^2 - 6 y + 71

3 [ x^2 - 4x + y^2 - 2y ] + 71        complete the square on x, y

3 [ x^2 -4x + 4 + y^2 -2y + 1 ]  + 71 - 12 - 3

3 [ ( x - 2)^2  + (y - 1]^2 ]  +  56

Q = (2, 1)

k  =  56

CORRECTED

                              A

12                               10                  18

                  Z                      Y

                                             8

         B                                       C

                            9.6

Since  CZ is a  bisector

Let  BZ = x  and  AZ =12 -x

So

AZ / AC  = BZ / BC

(12 -x) / 18 = x / 9.6

9.6 ( 12 -x)  = 18 x

115.2  - 9.6x = 18x

(18 + 9.6)x = 115.2

x  =115.2  / ( 18 + 9.6)   =  96/23   =  BZ  ≈ 4.17

Here :

https://web2.0calc.com/questions/please-help-thank-you-if-you-try_2#r1