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There are 6+4+8 buttons = 18, 4 are blue, thus the probability of getting a blue is 4/18  = 2/9. If we don't want a blue, 1 - 2/9 = 7/9 which is our probability.

If you don't understand anything feel free to ask.

If it wasn't AoPS, you wouldn't have the red slashes and brackets.

They are around all latex code.

If you want to try to act like you didn't know, it's not working.

For your information, AoPS can refer to

Art of Problem Solving

Artic Offshore Patrol Ship

Automated Office Protocol Standard

In case you were wondering, I was refering to the Art of Problem Solving.

What's AoPS

please fix your latex, and stop cheating on your AoPS homework.

"Find the largest positive integer n such that (n^2 + 7n + 12)/(n^2 - n - 12) is an integer."

have you got the link plz

sorry was this today

Geno answered this question recently.

If you aren't going to listen to Geno and disregard his/her solution, then you are plain disrespectful and you shouldn't even ask us for help if you aren't going to listen to the answer.

Since OB is also the radius, and is 7, the radius is 7, so OA is also the length of radius. \(\overline{OA} = 7\). 

\(\overline{OP} = \overline{OB} + \overline{BP} = 7 + 18 = 25.\)

Lastly since \(\triangle OAP\) is probably right, so.. using the pythogoream theroem, the hypothenuse \(\overline{OP}^2 = \overline{OA} ^2 + \overline{AP}^2 = 25^2 = 7^2 + \overline{AP}^2 \)

Thus,

\(\overline{AP}^2 = 25^2 - 7^2 = 625 - 49 = 576\)

SO \(\overline{AP} = \sqrt{576} = 24\)

So those are your answers!
 

If you dont' understand anything feel free to ask!

The highest point is 12cm, and the other part is 8, so the difference between the lower and higher prisms is 4.

So I'm not going to split this apart and instead I'm going to actually just calculate the total volume and subtract the part that's not filled.

So the "dug out" portion of the prism has a volume of 4 * 6 * 10 = 240. The total, including that, is 15*12 * 6 = 1080.

We counted the 240, so 1080-240 = 840

If you dont understand anything feel free to ask.

As follows:

I have answered this in the other thread. Good job fixing the numbers.

So we have an item price x, and 30% off means the price is 0.7x, and another 20% off is 0.7 * 0.8x = 0.56x.

So I'm pretty sure it's 56 percent, so 56-50 = 6%.

If x is 100, the first discount makes it 70, and the second 0.8 * 70 = 56.

If you don't understand anything feel free to ask!

thank you!

Again. I can not see your numbers.

So 300 / 360 = 5/6. The arc thus is 1/6 of the circumference.

The circumference is 8^2 π = 64π. 1/6 of that is 64π/6 = 32π/3.

If you don't understand anything feel free to ask!

When you speak of 'hypotenuse'   and 'legs' you are describing a right triangle .....

You can work this problem without calculus (but it's very messy!):

 1 + (2/x) + (3/x²) + (4/x³) + (5/x⁴) + ...

=  1 + (1/x + 1/x) +  (1/x² + 2/x²) + (1/x³ + 3/x³) + ...

=  [1 + 1/x + 1/x² + 1/x³ + ... ]  +  [1/x + 2/x² + 3/x³ + 4/x⁴ + ...]

=  [1 + 1/x + 1/x² + 1/x³ + ... ]  +  [1/x + (1/x² + 1/x²) + (1/x³ + 2/x³)+ (1/x⁴ + 3/x⁴) + ...]

=  [1 + 1/x + 1/x² + 1/x³ + ... ]  +  [1/x + 1/x² + 1/x³ + 1/x⁴ + ...]  +  [1/x² + 2/x³ + 3/x⁴ + ...]

=  [Red Part] + [Blue Part] + [1/x² + (1/x³ + 1/x³) + (1/x⁴ + 2/x⁴) + (1/x⁵ + 3/x⁵) + ...]

=  [Red Part] + [Blue Part] + [1/x² + 1/x³ + 1/x⁴ + 1/x⁵ + ...] + [1/x³ + 2/x⁴ + 3/x⁵ + ...]

 Etc.

Now, let's see what we have:

    [1 + 1/x + 1/x² + 1/x³ + ... ] 

+  [1/x + 1/x² + 1/x³ + 1/x⁴ + ...]  =  (1/x)[1 + 1/x + 1/x² + 1/x³ + 1/x⁴ + ...]

+  [1/x² + 1/x³ + 1/x⁴ + 1/x⁵ + ...]  =  (1/x²)[1 + 1/x + 1/x² + 1/x³ + ...]

+ ...

=

   (1)[1 + 1/x + 1/x² + 1/x³ + ... ] 

   +  (1/x)[1 + 1/x + 1/x² + 1/x³ + 1/x⁴ + ...]

       +  (1/x²)[1 + 1/x + 1/x² + 1/x³ + ...]

           + (1/x³))[1 + 1/x + 1/x² + 1/x³ + ...]

               + ...

=  [1 + 1/x + 1/x² + 1/x³ + ...] · [1 + 1/x + 1/x² + 1/x³ + ...]

=  [ 1 / (1 - 1/x) ] · [ 1 / (1 - 1/x) ] 

=  [ x / (x - 1) ] · [ x / (x - 1) ]

=  [ x / (x - 1) ] ²

For this problem:  (1998 / 1997 )²

Runner A   runs faster than B .....   every lap A gets closer to B

   Each lap takes A 5 seconds less than B        60 seconds/ 5 seconds/lap = 12 laps

   in 12 laps  B has taken  12 x 65 = 780 seconds        in 780 seconds  A will have run  780 / 60 = 13 laps    and will be at the same spot as B

   780 seconds

Please fix your latex, and add the image.

What's the constant term of the expansion of (do + not + cheat + on + your) * (A*o*P*S + Homework) ?