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There is no methodical reasoning behind five-digit perfect square palindromes. It's a matter of squaring all the numbers from 100 to 316.

(317² = 100489)

101² = 10201

111² = 12321

121² = 14621

202² = 40804

212² = 44944

264²  = 69696

307² = 94249

For four-digit perfect square plaindromes... there are none!

So, your answer would be the 7 five-digit perfect square palindromes that we stated above.

.........

 GAC  will form a right triangle

Let  GC  =  the  side of the cube  = S

Then AC  will = S√2

And  GA  = S√3

So.....GA  is the hypotenuse   and   AC  and  GC will form the legs

So

sin GAC   =   GC / GA   =    S   / [ S√3 ]  =  1  /√ 3 

[ To find the angular measure  use  arcsin (1/ √3)  ≈  35.26° ]

So do you know who made the website?

We can find the  direction vector, AB  =  s ,  of the line thusly

( 2 - 1, 3 - 1, 4 - 2)   =  ( 1, 2,  2)  = s

Next....find the vector  QA  =  ( 1 - 3, 1  - - 7, 2 - - 1)  =  ( -2, 8, 3)

Next.....form the cross product  of    QA  x s  =

i      j       k       i      j

-2   8      3      - 2   8     =

1    2      2      1     2

[ i * 8 * 2  + j * 3 * 1 + k * -2 * 2]  -  [ k * 8 * 1 + i * 3 * 2  + j * - 2 * 2 ]  =

[ 16i  + 3j - 4k] - [ 8k + 6i - 4j]  =

10i + 7j  - 12 k   

The distance   =

Length  of   [QA x s]

_________________   =

Length  of   [ s ]

√ [ 10^2 + 7^2 + (-12)^2  ]                 √ 293         √ 293

_______________________   =     _______  =  ______  

√ [ 1^2 + 2^2 + 2^2 ]                           √9                 3

So  d  = 293

How did you figure that out guest!

9.54 - 4.59 =4.95

If a, b, or c = 0, then there is 15 reversable ordered pairs for any of the two other variables adding up to 30. 

( a, b ) = 30

( b, c ) = 30

( a, c ) = 30 

15 * 3 = 45. Since the ordered pairs are reversable between two variables, 45 * 2 = 90 

Similarly, if a, b, or c = 1, then there is 14 reversable ordered pairs for any of the two variables adding up to 29. 

14 * 3 * 2 = 84 

...

I advise to continue on with this method.

I didn't do this by myself.  I found the answer on the internet. 

To guarantee a number is a perfect square, cube, 4^th or 5^th power, we need to raise the number to the power of 3 × 4 × 5 = 60. We also want the number to be divisible by 2, 3, 4 and 5. If we raise 2 × 3 × 5 = 30 to any power, 2 or greater, then it will be divisible by these integers. So, 30⁶⁰ would satisfy our conditions. The least possible value of a + b is therefore 30 + 60 = 90.      

Why does it say two replies but I am the only reply

YIKES!    I had no inclination....or know-how .....to answer THIS one !

Nice , Melody ! 

it is because of global warming, but trees will help us  

Yes, because the more trees there are, the more shade we can get in. 

.

Hi Sketchy,

I have used the same logic as the other person but i have added a few extra words.

So if you can understand.

Feel free to ask questions or comment.

The least positive integer that is divisible by 2,3,4 and 5 and is also a perfect square, perfect cube, 4th power, and 5th power, can be written in the form a^b for positive integers a and b.

What is the least possible value of a+b?

The answer is 90

The number is a^b where  a   is the smallest number that will work.

For instance  4^3 is not ok because 4 = 2^2 so    4^3 can be written as 2^6

This is not worded well but more exact wording would probably just add to confusion anyway.

a^b is a square number so b is a multiple of 2

a^b is a cubic number so b is a multiple of 3

a^b is a power of 5 so b is a multiple of 5

a^b is a power of 4 so b is a multiple of 4 but it is already a multiple of 2 so we just need one more 2

That is

b=2*3*5*2 = 60      It could be bigger but that is the smallest it can be.

Now what number to the power of 60 is a multiple of 2,3,4 and 5

2^60 is not a multiple of 3 or  5 but it is a multiple of 2 and 4

(2*3)^60 is not a muliple of 5 but it is a muliple of 2,3 and 4

(2*3*5)^60 is a muliple of 2 and 3 and 4 and 5. 

so the smallest values of a and b are  30 and 60 respectively

a+b = 30+60 =90

How did you get 30?
 

What are the steps involved?

The distance is sqrt(30)/3, so d = 30.

I bet they are looking for you to use the Pythagorean Theorem. Since the random number will be less than 1, you can assume 1 is the largest side length. For obtuse triangles, a2+b2

now pi/4*4=4pi/4=pi.

So your answer is pi.

Chris you have answered over me for no good reason AND you left out the bit that would cause the most problem.

"Why is the length of the square 4 units and what has a triangle got to do with a square."

If this kid sees your  answer and wants to learn from it at all, which is quite unlikely, those are the questions they would be asking themselves.

I do admit my answer was very basic but it at least gave a hint on how to find the length of the triangle legs.

If the child wanted more help he/she should have, and probably would have asked.

wow so much easier

As an addendum, note  when we have this

x^2        y^2

___  +  ____  =     1          ( A )

80          80

c  = √[a^2  - b^2 ]  =   √[ 80^2 - 80^2 ]  =  √0  =  0

So

c  / a    =     0  / √80  =  0

But...notice that  if  we   mutiplied    ( A)   through by   80  we get

x^2  + y^2  = 80

Which is a circle centered at the origin with a radius  of √80

So.....the closer that  e  is to   0....the more circular

And the closer that e  is to 1, the more elongated

Very nice,  KS!!!!

Looks like you might develop this hobby into something more.....  

Here : https://web2.0calc.com/questions/please-help_34243

We found that the normal vector to the plane containing A, B , C  was

(-1, - 3 , - 1)

The dot product  of  any vector  parallel  to the plane and the normal vector to the plane will  =  0

So  using  ( 1, 2 , x)     we have that

1(-1) + 2(-3)  + x (-1)  =  0

-1  - 6  -  x  =  0

-7  - x  =  0

-7  = x

See here :  https://web2.0calc.com/questions/please-help_34243

The equation of the plane was found to be   :

1x + 3y +1z  - 4  = 0

So we can write

1x  + 3y  +  1z  =  4        multiplying this through by 2  we get that

2x + 6y + 2z  =  8

(a, b , c)   =  ( 2, 6 , 2)

e  =  c  /  a

c  =  √[ a^2  - b^2 ]  =   √[80 - 16]  = √64  =  8

a  =  √80  =  4√5

So

e   =    8   /  [ 4√5]   =   2 / √5    ≈  .89

If  the eccentricity  is   0.....we have a circle...so.....the closer  that the eccentricity  is to 1,  the  more elongated it is 

So....this ellipse  is  more elongated than circular

See here to confirm  this, Jenny  :  https://www.desmos.com/calculator/sij3lqn4ts