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Sure, I can help you, but maybe be a little more patient next time. Answerers need to take time to solve the problem.

I'm not so sure about my hint, (and I'm not too confindent in my answer) but hopefully it helps you think. For your second expression to be the smallest possible value, the values of the first equation has to have EVERY single value to be the smallest possible. Sure, you could set one value to 1/200000, but then the other values need to be larger. Using number sense, we can see that the most "logical" solution for a, b, c, and d is 1/4, 1/4, 1/4, and 1/4. Notice a pattern? That is right, they are all the same. Adding the squares up will give us (yet again!) 1/4.

Maybe wrong link???

oops, sniped by guest!

Can I have help?

I checked the link, and the quesition appears to be answered. Can you check again, maybe?

it has

Since all sides are integers, we can use pythagorean tripples to help solve this. If you dont know what that is, you can look here:

https://www.mathsisfun.com/pythagorean_triples.html (in general, mathisfun.com is a great place to search up the definition of a math term, as they give many cool diagrams and their wording is simple and easy to understand)

Anyways, we see that since BA is 3, and all sides are integers, CB has to be 4, and CA has to be 5 (3-4-5 triangle). From here, we see that since CA is the shorter leg of the blue triangle, CD has to be 12, and DA has to be 13 (5-12-13 triangle). And finally, the green triangle. Now, although DA looks like the longer leg, the only other pythagorean triple that contains 13 is a 13-84-85 triangle. So, here is a great example on why you should always use geometric intuition, but never assume. Therefore, the length of AE is 85.

It's not all integers from 1 to 123456789. it's all integers whose digits increase from right to left

Let's make the DIAMETER of the larger circle   1

   the diagonal of the square is thus  1

      using pythag theorem the sides of the square are then  sqrt2 / 2

         sqrt2/2 is also the DIAMETER of the smaller circle

pi r_s^2 / pi r_l^2          (s = amall   l = large)

r_s ² / r_l²   =   (sqrt2/4)^2 / (1/2)^2 =   2/16  / 1/4  = 8/16 = 1/2     =     ratio:     area small / area large

thank you guest for your effort and time, but I think that (2,3) may be wrong. I don't know what is wrong with it, since I think that the reasoning is quite clear, but I will take a few moments to see what could be wrong with it.

no problem!!!

Thx!

Expanding:

8 cbrt125    + 4 cbrt 50  +  2 cbrt 20      -   4 cbrt 50 - 2 cbrt20 - cbrt 8  =

8(5) - 2   = 38

The figure on the left:  

Because the lines are parallel,

   (5y - 4) + (3y)  =  180^o   --->   8y - 4  =  180^o   --->   8y  =  184^o   --->   y  =  23^o  

   3y  =  2x + 13   --->   3(23)  =  2x + 13   --->   69  =  2x + 13   --->   56  =  2x    --->   28^o  =  x

The figure on the right:

Because the lines are parallel,

   3x + 2y + 4y  =  180^o   --->   3x + 6y  =  180^o  

Because they are in a triangle,

   2y + 4y + 5x - 20^o  =  180^o   --->   6y + 5x  =  200^o 

Solving these equations:  6y + 3x  =  180^o 

                                         6y + 5x  =  200^o

Subtracting:                            -2x  =  -20^o

And, now it's your turn ...

Knowing that  (n + 1)!  =  (n + 1)·n!:

2(n + 1)!  +  6n!  =  3(n + 1)!

2·(n + 1)·n!  +  6·n!  =  3·(n + 1)·n!

Divide both sides by  n!:

2(n + 1)  +  6  =  3(n + 1)

    2n + 2 + 6  =  3n + 3

          2n + 8  =  3n + 3

                  8  =  n + 3

                  5  =  n

1.5 hr on freeway     rate = 3x

   .5 hr on mt road     rate = x

rate * time = distance

1.5 (3x)    +   .5 x     = 110        solve for x  the mt road speed

If you want help, maybe you could find a coding forum for that? I only know Python, but I'm sure there are forums for questions like those, like cplusplus.com. I haven't really dug into that site yet, but it seems like it could help.

Remeber what z is! Z stands for a complex number in the form a+bi, where a and b are real, and i is an imaginary number. So, we can restate the question like:

(a+bi)^2=2i.

(a+bi)^2 can be expanded into a^2+2abi-b^2 (becase the sqaure of i is -1).

Therefore, a^2+2abi-b^2=2i. For this to be true, we see that the only way is for a^2 and b^2 to cancel out. Along with that, 2ab must equal to 2. Therefore, the options for a and b are 1 and -1. 

Using combinations, we see that the complex numbers that satisfy this equation is: 1+i, -1+i, 1-i, -1-i.

not correct but thanks for trying :(

ughhhhhhhh crapppppppppp i ant figiore it outttttttttttttt ****

Draw a number line

put a at zero

put b  at 2

put c at 7

put d at 14

now you should be able to work out the ratio easily....

Oops!  Quite right.  I messed up that detail.  The method still holds but Firebolt will need to go through it correcting the numerics.

f1.f2.f3. = -3 ?

Use the formula for an AP:

Sum =[First term + Last term] / 2 x  Number of terms

Sum =[1002 + 10000] / 2 x  4500 =24,754,500

Like so:

Make sure you check this carefully to see if I've made any silly mistakes!