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Don's apples = D
Alan's apples = A
D = A * 1.666

A - 30 = (D -30)/5

A = 54
D = 90

Alan had 54 apples at first

x = 1 - x + x^2 - x^3 + x^4       rearrange  as

1  - 2x  +  x^2  - x^3   +  x^4  =   0

1   is   a  root.......using  synthetic  division  to  find  the remaining  polynomial

1  [  1     -1      1      -2         1    ]

               1      0       1        - 1

     _________________________

       1      0      1    -  1          0

The  remaining  polynomial  is

x^3  +  x    -   1

Set  to   0

x^3  + x   -  1     =   0                no  easy  algebraic  way  to  solve   this

WolframAlpha  shows  the  other  real root  as   x  ≈  .68233 

Find  what   ????

The  coefficient  of  the  x^3   term  is  given  by

C (7, 4)   (x)^3 ( 2sqrt 5)^4   =

35  x^3   ( 16 * 25)   =

14000 x^3

You  have already  taken  all  the aces,  2's and  3's  from  the  deck

There  is   0  probabilty   that  you   can  draw  an ace ,  2  and    3

The  center   of   the  circle  is  the  midpoint    =   (  [ 3 -1 ]  / 2  ,  [ 12  + 2 ]  / 2 )   =  (1, 7)

The  square  of  the  radius   =   ( 3 -  1)^2  +  (  12 - 7)^2   =  2^2  + 5^2   =  29

The  area  of  the  circle   =   29  pi  units^2  

No of chicken wings = x .

No of sliders      = y

75x + 250y = 800

              2x=y.

If    1  -  sqrt 2    and   2  +  sqrt 7    are  roots, then  so   are  the   conjugates  1  + sqrt 2   and  2  - sqrt 7

We  just  need  to  simplify  this

( x -  (1 - sqrt 2) )  ( x  -  ( 1  +  sqrt 2)) (  x - (2  + sqrt 7)) ( x - ( 2  -  sqrt 7))    =

x^4 - 6 x^3 + 4 x^2 + 10 x + 3

Answer=
 three point shots = 7

  free throws = 2

Analysis = Suppose he made x three point shots, and y free throws 

                It can be know from the title.

                 { x + y = p        { x = 7

                 {  3x + y = 23 =>    {  y = 2

Wow!   It DID take over 5 minutes to load !   

I think to be a multiple of three one of the two numbers must also be multiple of 3

33 numbers are mutilples of 3 between  1 -100

  the other number can be any of the 100

33 x 100= 3300   ways  if at least ONE of the two chose a 3 multiple

     (so there are 3300 pairs of numbers that will produse a number that is a multiple of 3......if the ORDER of the numbers does not matter....i.e. it does not matter who chose which of the pair.....which is what I expect your questioner wants for an answer)

Looks   like  EP  signed  out....so  I'll   answer

The  points   (  -1,1)    and  (-3, - 6)   are  on  the  graph

Slope   = (-6  -1)  / ( -3 - -1)  =  -7 / -2   =   7/2

Using  either point,  we  have

y  =  (7/2) ( x - -1)   +  1

y = (5/2)x +  7/2  + 1 

y = (7/2)x  +  9/2      multiply  through by  2

2y  =  7x   +  9         rearrange  as

7x  -  2y  =   -  9

No, EP........you  need  to   clairvoyant, too  !!!!

Actually....the  graph is  just slow-loading....

Hahahaha...

Let   the point   we are  looking for  be   ( m ,  (1/3)m^2 -9)

We  can minimize  the  square of  the  distance formula

D^2   =  (m)^2   + [ (1/3) m^2  - 9 ]^2

D^2   = m^2  +  (1/9)m^4  - 6m^2  +  81

D^2  =  (1/9)m^4  -  5m^2  +  81

Take  the  derivative  of  the functtion  and  set  to  0

D' ^2   =  (4/9)m^3  - 10m    = 0

(4/9)m^3  - 10m =  0

m   [   ( 4//9)m^2  -  10  ]   = 0

The  second factor  dives  us  what we need

(4/9)m^2   -  10   =  0

(4/9)m^2 = 10

m^2  =  45/2

We  have  two   possible  values   m   = sqrt (45/2)     or  m   =  -sqrt (45/2)

Either  will  give  us  D^2    =  a^2    =  (1/9)(sqrt (45/2))^4 -5 (sqrt  (45/2 ))^2  +  81  =   99/4

See  the  graph here  :  https://www.desmos.com/calculator/pleksd640o

a = 5      b = 6      c = 8

s = 1/2(a+b+c)

A = sqrt[s(s-a)(s-b)(s-c)]

r = A/s

The day in 1988 on which the year was 214.3 days old is:    

Adding up     Days   Month      Running Total  

                      31      January          31

                      29      February        60 

                      31      March             91  

                      30      April              121  

                      31      May               152  

                      30      June              182 

                      31      July                213      so the 214.3th day = August 2   

Kinda need the graph.....

Sorry if i wasn't thorough enough with my question it needs proof of it converging or diverging based on the n th number in the row by using some of the test of convergence( comparison test, Limit test,quotient test Dalamber criterion, root test Cauchy criterion) im just not sure where to start.

Hope you see this im new to the site and not sure if im responding correctly.