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The question is ambiguous.  Does it mean

1.Look for a value of k only using (2a-b,a); then look for a separate value of k using only (2a_b+2,a); then look for a separate value of k using (a+2b+2,b)

or

2. Look for values of k using one of the pairs selected at random at each stage

or

3. Something else?

If 1, then there is only one value of k, namely 2013. (the patterns become clear if you just work out a few of the initial stages for each expression).

Please do not post AoPS questions on here.

You are welcome.

Maybe you are a bit lucky too.

I do not remember seeing anyone else answer Excel questions on here. 

(Although there must be a handful of people here who could)

Is that all you want?  Did you want to know how to round it off to the nearest degree, or anything like that?

Also you could include pi in the equation.    Just replace B5 with   PI()        that would be more straight forward.

2   .√ ( log ₃ x⁹)  -  4log ₉ √(3x)  = 1

√ (log ₃ x⁹)  -  log ₉  √(3x)⁴    =1

√(log ₃x ⁹) -   log ₉ (3^2  x²)   =  1

√ (log ₃ x⁹) - log ₉ 3^2   -   log ₉ x²   =  1

√(9log ₃ x)  -  log ₉ 9  -  log ₉ x² =  1

√(9log ₃ x)  -  1  -  log ₉ x² =  1

√(9log ₃ x)    - 2 log ₉ x =  2

3 ( log 3  / log x)^(1/2)  -  2 (log x / log 9 )  =  2

3 ( log 3 / log x)^(1/2)   - 2  (log x  / log 3^2)   =2

3( log 3 / log x)^(1/2)   -  2  (log x /(2 log 3) )    = 2

3 ( log3/ log x)^(1/2)  - log x /log 3     = 2

3  (log 3/logx)^(1/2)    -  1 / (log 3/logx)    = 2

Let  (log 3/log x)^(1/2)   =  a

Then   log3/logx   =  a^2

So we have

3a   - 1/a^2   = 2        multiply through by a^2

3a^3   -1  =2a^2

3a^3  - 2a^2   = 1       and it's  clear that  this will be true when a  = 1

So      (log 3 / log x)^(1/2)    =1          square both sides

log 3 / log x  =  1

x  = 3

5.  We can write

(1/3)  ( log x / log 2)   + (log x /log 2)^(1/3)   =  4/3

Let Logx /log 2   =  a   

And we have

(1/3)a +  a^(1/3)   = 4/3      rearrange as

a^(1/3)    =  4/3  -  (1/3)a

a^(1/3)  =  (4 - a)   / 3        cube both sides

a  =   (4 - a )^3   /  27    

Note that this is true when  a =  1

So  a =    log x / log2   = 1      →    x  = 2

Here's the answer to the first one

[10 C 3] * 5^7 / 6^10 ==0.155

Thank you so much.
Good thing I met this site.

I do have an answer for  question1 

 let A=3^x   and   B=2^x

Do the substitution and then solve as a quadratic.

Where did 144*pi come from???

------------------------------------------------

64% of 288 = 184.32

V = 4/3*pi*r³                              

288pi = 4/3*r³                        184.32pi = 4/3*r³

288pi*3 / 4 = r³                       184.32pi*3 / 4 = r³

r³ = 216                                   r³ = 3456/25

r = 6                                         r = 5.170643256

Ratio             r : r =  r = 5.170643256 / 6      ==>     0.861773876  ≈ 431/500

Any help?

btw please show how you did it, i want to learn how to do this type of log because im kinda interested

This can be solved with Heron's formula, which states that the area of a triangle with side lengths \(a, b\) and \(c\) and semiperimeter \(s\) has area\(\sqrt{s(s-a)(s-b)(s-c)}\). Applying this we have the area of the triangle is \(\sqrt{121\cdot 231\cdot 127\cdot 126} \approx 15926\). I hope this was helpful! :D

PQ = 2             2² / 4 = 1           area of PQR = 1

Go to your Art of Problem Solving message board to get help with this question.

Do not cheat and defeat the whole purpose of the AoPS learning site by asking people here to do it for you.

A forest covers

73,000

acres. A survey finds that

0.8​%

of the forest is​ old-growth trees. How many acres of​ old-growth trees are​ there?

0.8% = 0.008  

            0.008 x 73,000 acres  =  584 acres  

_.

Its not correct

Can you explain further, because I have to show my working?

There are 422 possible values of k.

Here is a lead: notice how the average of entire set is only $\frac {n+2}2~or~\frac n2+1$

You did 13 hours of homework.

I messed up and found my err. I have edited in the corrections. My apologies.

In red is what I have corrected.