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Thank you so much Aven!

Note that the left-hand side is divisible by 3. Therefore, since 3 divides 6, the left-hand side will be equivalent to a multiple of 3. Therefore, if x were to be 1, 2, 4, or 5, then the given congruence would have no solutions. 4 values.

:D

Good job, Aven John!

The prime numbers are 2, 3, and 5.

Sets with one number: 3choose1=3

Sets with two numbers: 3choose2=3

Sets with three numbers: 3choose3=1

Total: 7

:D

So inversely proportional means that \(x^2y\) is always the same, and \(6\times6=36, 36\times27=972\), \(972/8=121.5\), so x is the square root of 121.5

i do aops TOO guest. stop trying to cheat your writing problem. btw the aops admins know when your cheating. ;)

AVEN JOHNNNNNNNNN

Answer: 60

Explanation:

We know f(1) = 4. Then we can find f(1+1) = f(2), because of the formula they give. Plugging in x=y=1 we get f(1+1)=f(1)+f(1)+1*1 --> f(2) = 4+4+1 --

f(2) = 9.

We can continue on until we get f(8). Letting x=y=2 we get f(2+2)=f(2)+f(2)+2*2 --> f(4) = 22. Finally, letting x=y=4 we get f(4+4)=f(4)+f(4)+4*4 --> f(8) = 60.

You are very welcome!

:P

Answer:

$972

Step-by-step explanation:

Ajax buys 20 shares of stock at $20.

The cost of 20 shares = 20 × $20 = $400

The shares increase in value by 20%

Now the cost of 20 shares = 400 + (20%×400)

                                            = 400 + 80

                                            = $480

Cost per share =   = $24

He sells 10 shares = $20 × 10 = $240

Now he has 10 share costs $240

After the shares decrease in value by 10%, he buys another 20 shares.

Cost per shares after decrease by 10% = 24 - (10%×24)

                                                                 = 24 - 2.40

                                                                 = $21.60

He buys 20 shares and his remaining 10 shares = 20 + 10 = 30 shares

Cost of 30 shares = 21.60 × 30 = $648

The shares then all increase by 50% = 648 + (50% × 648)

                                                             = 648 + (0.5 × 648)

                                                             = 648 + 324

                                                             = $972

The total value of the 30 shares that Ajax own is $972

ok tysm!

Yes because 0 to 9 has 10 is 0,1,2,3,4,5,6,7,8,9 so 10

6^2*27=972

8*x^2=972

x=sqrt(121.5)

We have that \(8=x^2+y^2=x^2+2xy+y^2-2xy=(x+y)^2-2xy=16-2xy\), therefore .\(xy=\frac{16-8}{2}=4\) Since \(x^3+y^3=(x+y)(x^2-xy+y^2)=(x+y)(x^2+y^2-xy)\), we can directly substitute in the numerical values for each algebraic expression. This gives us \(x^3+y^3=(4)(8-4)=\boxed{16}\).

That's actually the answer I see in the answer key but would I put 10 as the denominator and not 9?

\(-0.3x+3=4.2\)

\(-0.3x=1.2 \)

\(x=-0.4 \)

-0.3x=1.2

x=-0.4

:D

(x+y)^2-x^2-y^2=xy=8

xy*(x+y)=32=x^2*y+x*y^2

(x+y)^3-3x^2*y-3y^2*x=x^3+y^3=-32

:D

The number of digits on each dial is 10 and they are independent of each

other (mutually exclusive) so the number of different combinations is ....

==> 10 x 10 x 10 x 10 = 10,000

So, the probability of guessing the combination is 1 / 10,000 = 0.0001

Very close, pay attention to how it does not specify the order in which the three spins have to happen. For example:

a) asks for red, blue, and yellow... but it does not have to be in that order. There is a 1/3 chance for each color to be landed on... so there is a 1/27 chance that in that order, will be the outcome.

But, since it could also be blue, red, and yellow or yellow, red, and blue. We have to account for that as well.

There are 3*2*1 different combinations. Therefore your answer is 6/27 or 2/9. (for part a)

See if you can do the rest :D..

---

Also, I can't say that I am always right, so if you think I made a mistake, just let me know :)

That's not correct because each term is greater than the one before it. Since there are 49 pairs, we have to subtract 49 to get 88 divide by 2 to get 44 and add the 49 back. 93.

:D

Actually, I just figured out what I did wrong, there are 13 terms, not 12... The correct answer is 5304.

Since we are taking exactly haf the terms, a_2 + a_4 + ... + a_98 = 137/2.

This looks accurate to me...

That's what I did and the answer key said it was wrong :/

a) 1/3*1/3*1/3=1/27

b) 1/3*1/3*2/3=2/27

c) 1/3*1/3*1/3=1/27

:D