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We can use the multiplying strategy for this problem : 

6 x 4 x 4 x 2 x 2  = 384 ways to sit 

1 + (sumfor(n, 1, 1000000, (5^ -n)*(1 + 2*n)))==converges to 15/8 ==1 + 7/8

If I log on the Martians might track me using the session cookie.

I think the Martians are the ones who helped WAT escape from the loony bin. 

GA

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https://latex.artofproblemsolving.com/texer/d/dagugvpy.png?time=1689715090631

The powers of i repeat in blocks of four:

i,i2,i3,i4=1,−1,−i,i

Therefore, the sum telescopes:

\(\begin{align*} i + 2i^2 + 3i^3 + 4i^4 + \dots + 64i^{64} &= i + 2(-1) + 3(-i) + 4i + \dots + 64i \\ &= (i - 2i + 3i - 4i + \dots + 64i) + (2 - 3 + 4 - \dots + 64) \\ &= 0 + (2 + 64) \cdot 32 \\ &= \boxed{496}. \end{align*}\)

If I give my brother 5 dollars, then we will have the same amount of money. If instead he gives me 25 dollars, then I'll have twice as much money as he will have. How much money does my brother currently have (in dollars)?   

Let M stand for My original amount  

Let B stand for Brother's original amount   

                                                             (M – 5)  =  (B + 5)   

                                                             (M + 25)  =  (2)(B – 25)  

                                                              M – 5  =  B + 5   gives us  M = B + 10    (1)  

                                                              M + 25  =  2B – 50                                 (2)  

Substitute the value of M from (1)   

into the equation (2)                             B + 10 + 25  =  2B – 50   

Combine like terms                                          85  =  B    

Brother has 85 dollars ... before they start swapping moneys.   

The problem doesn't ask for My original amount but it's B + 10 = 95  

Check answer  

If I give my brother 5, then I'll have 90 and so will he, so we have the same.  

If my brother gives me 25, then I'll have 120 and my brother will have 60.  

_.

Will and Grace are canoeing on a lake.  Will rows at 50 meters per minute and Grace rows at 20 meters per minute. Will starts rowing at 2 p.m. from the west end of the lake, and Grace starts rowing from the east end of the lake at 2 p.m. If they always row directly towards each other, and the lake is 3800 meters across from the west side of the lake to the east side, at what time will the two meet?  

Will's rate is 50 m/min.  

Grace's rate is 20 m/min.  

They're rowing toward each other,  

so their rate of closure is 70 m/min.  

The lake is 3800 meters across.  

The time it will take them to meet is given by  

                                                                                    Distance  

                                                                            T  =  –––––––  

                                                                                       Rate  

                                                                                    3800 m  

                                                                            T  =  –––––––  

                                                                                    70 m/min  

The time it takes them to meet                             T  =  54.29 min  

Note that (0.29 min • 60 sec/min)  =  17 seconds

The time on the clock is                                             2:54:17 pm 

_.

         x   y  z    Maximum Value

1  = (3, 1, 2) = 0.25
2  = (2, 1, 3) = 0.25
3  = (6, 2, 4) = 0.25
4  = (4, 2, 6) = 0.25
5  = (9, 3, 6) = 0.25
6  = (6, 3, 9) = 0.25
 

Why hello there homework cheater!

The answer is 52

GA

This is correct.

GA

bruh

okay

This spammer has been posting the same nonsensical question over and over. Best not to engage.

We can write the expression as [\frac{xyz(x + y + z)}{(x + y)^2 (y + z)^2} = \frac{xy(z(x + y) + 1)}{(x + y)^2}.]By AM-GM, [xy(z(x + y) + 1) \le \sqrt{xy \cdot xy \cdot (z(x + y) + 1)^2} = \sqrt{(xy)(z(x + y))^2 (1 + 1/z)^2} = \sqrt{(x + y)^2 (y + z)^2},]so [\frac{xy(z(x + y) + 1)}{(x + y)^2} \le 1.]

Equality occurs when z=yx+y​, so the maximum value is 1​.

I think that you mean that when six juniors drop the class and one senoir enters the class, the number of juniors to the number of seniors because 2:3. 

We know that 3:2 is 6:4 is 9:6 is 12:8... We can check one by one whether or not the answer is correct. 3:2 does not work, neither does 6:4, or 9:6, but 12:8 works. (12-6):(8+1) = 6:9, or 2:3. Therefore, the answer to this problem is 12:8(make sure to read the top to make sure that you question is the same as what I think it is :)). 

That's not correct.