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By computer and using a CAS on that computer.

What is the formula for the surface area of sphere? If you do not know it, how can you find it?

I can solve in terms of x:

y=1/(-x-1) and z->(-x-1)/x

So, there are infinite number of solutions.

You should post it.

That will not be good enough:

Here are 100 solutions, I am sure there are lots more...

{{x -> -(5645/10002), y -> -(10002/4357), z -> 4357/5645}, {x -> 1334,

y -> -(1/1335), z -> -(1335/1334)}, {x -> -(3199/5001),

y -> -(5001/1802), z -> 1802/3199}, {x -> 8015, y -> -(1/8016),

z -> -(8016/8015)}, {x -> 7170, y -> -(1/7171),

z -> -(7171/7170)}, {x -> 16, y -> -(1/17),

z -> -(17/16)}, {x -> 1036, y -> -(1/1037),

z -> -(1037/1036)}, {x -> -342, y -> 1/341,

z -> -(341/342)}, {x -> 7533, y -> -(1/7534),

z -> -(7534/7533)}, {x -> -8781, y -> 1/8780,

z -> -(8780/8781)}, {x -> 684, y -> -(1/685),

z -> -(685/684)}, {x -> -(2661/3334), y -> -(3334/673),

z -> 673/2661}, {x -> -1960, y -> 1/1959,

z -> -(1959/1960)}, {x -> -(1019/5001), y -> -(5001/3982),

z -> 3982/1019}, {x -> 7750, y -> -(1/7751),

z -> -(7751/7750)}, {x -> 8643, y -> -(1/8644),

z -> -(8644/8643)}, {x -> -(26/1667), y -> -(1667/1641),

z -> 1641/26}, {x -> 9659, y -> -(1/9660),

z -> -(9660/9659)}, {x -> 2765, y -> -(1/2766),

z -> -(2766/2765)}, {x -> 7307, y -> -(1/7308),

z -> -(7308/7307)}, {x -> -5606, y -> 1/5605,

z -> -(5605/5606)}, {x -> -3266, y -> 1/3265,

z -> -(3265/3266)}, {x -> -(1037/5001), y -> -(5001/3964),

z -> 3964/1037}, {x -> 7906, y -> -(1/7907),

z -> -(7907/7906)}, {x -> -(1640/1667), y -> -(1667/27),

z -> 27/1640}, {x -> 7264, y -> -(1/7265),

z -> -(7265/7264)}, {x -> -9128, y -> 1/9127,

z -> -(9127/9128)}, {x -> -4750, y -> 1/4749,

z -> -(4749/4750)}, {x -> -847, y -> 1/846,

z -> -(846/847)}, {x -> -(8881/10002), y -> -(10002/1121),

z -> 1121/8881}, {x -> 5682, y -> -(1/5683),

z -> -(5683/5682)}, {x -> -8386, y -> 1/8385,

z -> -(8385/8386)}, {x -> -4467, y -> 1/4466,

z -> -(4466/4467)}, {x -> -7428, y -> 1/7427,

z -> -(7427/7428)}, {x -> -7210, y -> 1/7209,

z -> -(7209/7210)}, {x -> -6297, y -> 1/6296,

z -> -(6296/6297)}, {x -> -(497/5001), y -> -(5001/4504),

z -> 4504/497}, {x -> -(3401/10002), y -> -(10002/6601),

z -> 6601/3401}, {x -> -1157, y -> 1/1156,

z -> -(1156/1157)}, {x -> -(8083/10002), y -> -(10002/1919),

z -> 1919/8083}, {x -> 3292, y -> -(1/3293),

z -> -(3293/3292)}, {x -> -(1300/1667), y -> -(1667/367),

z -> 367/1300}, {x -> 1605, y -> -(1/1606),

z -> -(1606/1605)}, {x -> 8865, y -> -(1/8866),

z -> -(8866/8865)}, {x -> -(1457/1667), y -> -(1667/210),

z -> 210/1457}, {x -> -(3991/5001), y -> -(5001/1010),

z -> 1010/3991}, {x -> -(2422/5001), y -> -(5001/2579),

z -> 2579/2422}, {x -> -5372, y -> 1/5371,

z -> -(5371/5372)}, {x -> -(113/5001), y -> -(5001/4888),

z -> 4888/113}, {x -> -(7949/10002), y -> -(10002/2053),

z -> 2053/7949}, {x -> -(893/5001), y -> -(5001/4108),

z -> 4108/893}, {x -> -9981, y -> 1/9980,

z -> -(9980/9981)}, {x -> -1136, y -> 1/1135,

z -> -(1135/1136)}, {x -> -4269, y -> 1/4268,

z -> -(4268/4269)}, {x -> 2015, y -> -(1/2016),

z -> -(2016/2015)}, {x -> -3587, y -> 1/3586,

z -> -(3586/3587)}, {x -> -(1423/5001), y -> -(5001/3578),

z -> 3578/1423}, {x -> -(2135/10002), y -> -(10002/7867),

z -> 7867/2135}, {x -> -4795, y -> 1/4794,

z -> -(4794/4795)}, {x -> -(3946/5001), y -> -(5001/1055),

z -> 1055/3946}, {x -> -(2761/5001), y -> -(5001/2240),

z -> 2240/2761}, {x -> -(1901/5001), y -> -(5001/3100),

z -> 3100/1901}, {x -> 7228, y -> -(1/7229),

z -> -(7229/7228)}, {x -> -2431, y -> 1/2430,

z -> -(2430/2431)}, {x -> -(489/1667), y -> -(1667/1178),

z -> 1178/489}, {x -> -(7021/10002), y -> -(10002/2981),

z -> 2981/7021}, {x -> -1413, y -> 1/1412,

z -> -(1412/1413)}, {x -> -(2090/5001), y -> -(5001/2911),

z -> 2911/2090}, {x -> 1041, y -> -(1/1042),

z -> -(1042/1041)}, {x -> -(2671/3334), y -> -(3334/663),

z -> 663/2671}, {x -> 9561, y -> -(1/9562),

z -> -(9562/9561)}, {x -> 9458, y -> -(1/9459),

z -> -(9459/9458)}, {x -> -188, y -> 1/187,

z -> -(187/188)}, {x -> -2658, y -> 1/2657,

z -> -(2657/2658)}, {x -> 2014, y -> -(1/2015),

z -> -(2015/2014)}, {x -> 8477, y -> -(1/8478),

z -> -(8478/8477)}, {x -> -6491, y -> 1/6490,

z -> -(6490/6491)}, {x -> -2633, y -> 1/2632,

z -> -(2632/2633)}, {x -> -5836, y -> 1/5835,

z -> -(5835/5836)}, {x -> -5822, y -> 1/5821,

z -> -(5821/5822)}, {x -> -(6427/10002), y -> -(10002/3575),

z -> 3575/6427}, {x -> -671, y -> 1/670,

z -> -(670/671)}, {x -> 1761, y -> -(1/1762),

z -> -(1762/1761)}, {x -> 6228, y -> -(1/6229),

z -> -(6229/6228)}, {x -> -5397, y -> 1/5396,

z -> -(5396/5397)}, {x -> -191, y -> 1/190,

z -> -(190/191)}, {x -> -(4235/10002), y -> -(10002/5767),

z -> 5767/4235}, {x -> 7467, y -> -(1/7468),

z -> -(7468/7467)}, {x -> -7228, y -> 1/7227,

z -> -(7227/7228)}, {x -> -5515, y -> 1/5514,

z -> -(5514/5515)}, {x -> 3703, y -> -(1/3704),

z -> -(3704/3703)}, {x -> 4663, y -> -(1/4664),

z -> -(4664/4663)}, {x -> -(9941/10002), y -> -(10002/61),

z -> 61/9941}, {x -> -4319, y -> 1/4318,

z -> -(4318/4319)}, {x -> -9316, y -> 1/9315,

z -> -(9315/9316)}, {x -> 7160, y -> -(1/7161),

z -> -(7161/7160)}, {x -> 3059, y -> -(1/3060),

z -> -(3060/3059)}, {x -> 2269, y -> -(1/2270),

z -> -(2270/2269)}, {x -> -3399, y -> 1/3398,

z -> -(3398/3399)}, {x -> 6726, y -> -(1/6727), z -> -(6727/6726)}}

So

Find all the triplets of distinct real numbers (x,y,z) for which:

will be difficult to do in a post. I could write the x,y relationships that would.

**bobbym**- Replies: 0

Hi;

https://www.youtube.com/watch?v=FWS5xfIaoeA

Look at him go! Now Lakeland is only 89 miles away from me... Could be the match of the century if that big fat out of shape monster wanders into my territory... Him with huge jaws, sharp claws and the strongest bite grip in nature and me with a napkin, plate and knife and fork. Ain't eatin a gator but I do not mind trying. Anyway, did you notice those imbeciles hiking along completely unaware that Godzilla ain't in Tokyo like those cheap movies conspire to make us believe. He lives right here in Florida enjoying the warm sun and an occasional imbecile strolling by.

You see the problem is in words like park. To a yank like myself, a park is a place with a couple of picnic tables, restrooms and water fountains. A place where Daddy can take his kids to see the ducks on the small pond in the center. Down here it has a totally different meaning. First time we got caught in what these people call a park we almost died.

Here parks have rattlers, cottonmouths, bears, feral pigs, cougars, panthers, boa constrictors, pythons and of course gators and crocs. To go into a park here you carry survival gear, you do not go around in shorts and sandals. Not unless you want to be part of the food chain.

"They were ready to jump to their feet?" Gators are faster than humans are. They use the word protected in the report perhaps to imply that those happy meals on two legs were perfectly safe. Sorry folks, the truth is the gators are protected by federal law, the dummies ain't.

And of course the news got it all wrong again. When we were in Myakka we saw a gator at least that big and not so out of shape. That size is not amazing, it is more common than they want anyone to know.

Hi;

You will need a bit more to get a solution. Is there a constraint you left out?

Hi;

You can divide by 2 or multiply by 1 / 2. Welcome to the forum.

Hi;

Welcome to the forum.

That would be a piece of luck. M exposes all my weaknesses.

Hi;

Thanks for the link. I understand how you want to count them say for ()()(). But what is the question you are asking?

I do see the formula you are conjecturing on page 5 of that pdf.

Hi;

I think I understand the problem now. I have notes on this bracketing problem somewhere. I am looking for them now.

Hi;

I do not understand about R(t), L(t), can you explain further.

The numbers you are suggesting are every other Catalan number.

Hi;

The solutions found in post #10 are correct but I am afraid their might be more that ht method I used there is missing. I hate using reasoning in math, that is why I dislike and mistrust the whole concept of proof. Trouble is, computation although more reliable can leave you hanging.

It is very desirable to get an analytic form. Can you do that, with the same rules still applying?

Hi chamywak;

The number they want next is

.Welcome to the forum.

Those were just thoughts, that might have a big hole in it.

There is undoubtedly a much shorter way to do this.

This will be the toughest baffler ever posed here.

Disclaimer: Do not try this problem and then drive or operate heavy machinery.

Okay, you see that side PQ is the square root of 3 and the equation of the ellipse is given below it.

The rules are rather extreme:

You may not use any CAS, that means Mathematica, Maxima, Wolfram Alpha, Maple, Matlab, Mupad, Octave, Pari, Yacas, Fricas, Magma.

You may not use any programming language, that means C, C++, Wolfram, Haskell, Basic, Pascal, Cobol, Fortran, Python or any other.

You may not use any mathematics that means calculus, group theory, algebra, trigonometry, geometry (Euclidean or any other), probability, any math theorem.

You can only use the tools of EM or you can try the force.

Okay, now for the question:

If we move P and Q along the circumference of that ellipse what is the sum of the largest area and the smallest area for triangle POQ? Of course the length of PQ must always remain constant at

A says) 2.

B says) That is not correct.

C says) 6.022 x 10^23

D says) 0

E says) I know the answer.

Yes, Length is the command that will tell you how long that list is. You can try this.

```
Select[Range[10000], PrimeQ[#^2 + 27] && PrimeQ[# + 3] &];
Total[%]
Length[%%]
```

Have a good lunch.

Hi;

For a functional approach:

`Select[Range[10000], PrimeQ[#^2 + 27] && PrimeQ[# + 3] &] // Total`

We can say that

Solving simultaneously:

{{x = -5, y = -3}, {x = -5, y = 3}, {x = 5, y = -3}, {x = 5, y = 3}}

Are the only 4 solutions and they all give n = 13.

Very good work!