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Product won't equal 6:  1 & 2;  1 & 3;  1 & 4; 1 & 5;  2 & 4;  2 & 5;  2 & 6;  3 & 4;  3 & 5;  3 & 6;  4 & 5;  4 & 6;  5 & 6

Product will equal 6:  1 & 6;  2 & 3

Just remember that you need not count duplicates; once you have counted  1 & 3  you don't count  3 & 1.

8  on top level will have paint  on at least two faces

4  on the  middle  level will have  paint on at least  two faces

8  on the  bottom level will have paint on at least two faces

So

8 + 4 + 8    =  20

Yeah!

y  =  ax + b     --->   the slope (a) represents the weekly deposit

                               the intercept (b) represents the initial deposit

Mark:         initial deposit = $2000   weekly deposit = $35      his equation:  y  =  35x + 2000

Elizabeth:  initial deposit = $2800   weekly deposit = $45      her equation:  y  =  45x + 2800

Elizabeth deposited $800 more than Mark.

Mark deposits $10 less each week than Elizabeth

Elizabeth deposits $45 for every $35 that Mark deposits; as a fraction, this is $45/$35  =  9/7, not 7/9

Choose  any  2  of  the  11  = 

C(11,2)   =   55 different pairs

I did  not  notice  that  it was a RT

However....the  Law of Cosines  will  always  work

A five-digit palindrome would be of the form ABCBA, where A, B, and C are digits (not necessarily distinct) from 0 to 9 (with A from 1 to 9). We have three possibilities for each of A, B, and C which gives us 3^3 possible 27 palindromes.

why do you need to use law of cosines?

The triangle is a right triangle, so can´t you use the normal cos(A)= adjacent over hypotenuse

thank you I was so confused on the two problems 

Points  on graph  (8,0)  (0, - 4)

Slope    [ -4 - 0]  / [ 0 - 8]  =  -4/ -8  =  1/2

Using either  point and the slope

y  = (1/2) ( x - 0) - 4

y = (1/2)x  - 4

Law  of Cosines

6^2  = 16^2   +  17.1^2  -  2 [ 16 * 17.1] cos B       rearrange  as

[ 6^2  - 16^2  - 17.1^2 ]  / [ -2 * 16 * 17.1 ]   = cos B

Take the  cosine inverse

arccos   ( [ 6^2  - 16^2  - 17.1^2 ]  / [ -2 * 16 * 17.1 ]  )  = B ≈ 20.54°

Likewise

[  16^2  -6^2 - 17.1^2 ]  / [ -2 * 6 * 17.1 ]   =  cos  A

arccos  ( [  16^2  -6^2 - 17.1^2 ]  / [ -2 * 6 * 17.1 ] )  =  A ≈   69.34°

We have  the points (0, - 6)  and (3, 6)  on the graph

Slope  =  [  6 - - 6] / [ 3 - 0 ]  =   12 / 3  =  4

Using the slope and either point we have  that

y = 4(x - 0) -  6

y  = 4x  -   6

I already did, I think. Check it out!

1,2 and  9

Here's what I get  :

11111          21112        91119    

11211         21212        91219

11911         21912        91919

12121        22122         92129

12221        22222         92229

12921        22922         92929

19191        29192          99199

19291       29292          99299   

19991       29992          99999

1/3 times 30

=30 divided by 3

=10

Adam runs  at 6mph  =  (6/60)*88  = 8.8 ft / s

Caroline  bikes  at  (11/60)* 88  =  242/15 ft/s

When Caroline  starts to ride, Adam  will have biked  (8.8)ft/s  * 600s  =  5280 ft

So....it will take  Caroline  (5280) / (242/15  -  8.8)   =  720  sec   to catch  Adam  = 12 min

50 + 12  =  62  min  after  this  she  passes  Benjamin =  time  that Benjamin has ridden

So   she has  biked  62 * 60 ( 242/15)  =  60016 ft  =  341/30  mi

So....Benjamin has run 18  - 341/30  = 199/30  mi in 62  min

So.....his  rate  is     (199/30)mi  ( 60/62)  ≈  6.42 mi/hr

Thankyou the answer was 45.5 and your explanation was great

I used to do AoPS forums, but it's more of chatting than actual problem solving.

I think some parts of your question is missing. Go here to find more.

https://web2.0calc.com/questions/help-please_7332