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Everyone was very helpful.

11,111,111,111,111,111,111 - 2,222,222,222

1 1 , 1 1 1 , 1 1 1 ,1 1 1 ,1 1 1 ,1 1 1 , 1 1 1

                                   - 2 , 222 , 222 , 222

___________________________________

11111111108888888889

Answer = 11111111108888888889
 

-\(tommarvoloriddle \)

11,111,111,111,111,111,111- 2,222,222,222 = 11111111108888888889

is there meant to be a picture for this problem? you can delete this problem if you wish, or you could send a link for the picture if there is meant to be one, because we do not know where a and b are.

\(2^{x+y}\,\cdot\,3^{x-y}\,\cdot\,6^{2x+2y}\ =\ 72\\~\\ 2^{x+y}\,\cdot\,3^{x-y}\,\cdot\,6^{2(x+y)}\ =\ 72\\~\\ 2^{x+y}\,\cdot\,3^{x-y}\,\cdot\,(6^2)^{x+y}\ =\ 72\\~\\ 2^{x+y}\,\cdot\,3^{x-y}\,\cdot\,36^{x+y}\ =\ 72\\~\\ 2^{x+y}\,\cdot\,36^{x+y}\,\cdot\,3^{x-y}\ =\ 72\\~\\ (2\,\cdot\,36)^{x+y}\,\cdot\,3^{x-y}\ =\ 72\\~\\ 72^{x+y}\,\cdot\,3^{x-y}\ =\ 72\)

At this point we can see that the equation will definitely be satisfied if

x + y  =  1    and

x – y  =  0

So    x  =  1/2    and    y = 1/2    is definitely a solution.

But that doesn't necessarily mean it is the only solution.

I don't know how to prove that it is the only rational solution, but it does seem to be.

Maybe someone else can explain why that is.

Go online to this page and learn how to solve it:

http://www.math.cmu.edu/~bkell/21110-2010s/extended-euclidean.html

Using Newton-Raphson method, it gives the following solution:

x = 1/2   and     y=1/2

No one?

Draw a line segment from  A  perpendicular to  DC  that meets  DC  at point  E , and

draw a line segment from  B  perpendicular to  DC  that meets  DC  at point  F , like this:

The sum of the interior angles in  △ADE  =  180°

45°  +  90°  +  m∠DAE   =   180°

m∠DAE   =   180° - 45° - 90°

m∠DAE   =   45°

△ADE  is an isosceles triangle and the sides opposite the base angles are congruent.

DE  =  AE

And if   DE = b   then   AE = b

By the Pythagorean Theorem,

DE² + AE²  =  (8√2)²

b² + b²  =  (8√2)²

2b²   =   128

b²   =   64

b   =   8

ABFE  is a rectangle so

BF  =  AE  =  8

EF  =  AB  =  4

By the Pythagorean Theorem,

BF² + FC²  =  10²

8² + FC²  =  10²

FC²  =  10² - 8²

FC²  =  36

FC  =  6

Now we know:

AB  =  4

So we can say  base₁ =  4

DE  =  8   and   EF  =  4   and   FC  =  6

So we can say  base₂  =  8 + 4 + 6  =  18

AE  =  8

So we can say  height  =  8

area of trapezoid  =  (1/2)(base₁ + base₂)(height)

area of trapezoid  =  (1/2)(4 + 18)(8)

area of trapezoid  =  88

If you have the 45-45-90 triangle memorized, you can immediately recognize that DE and AE  must be 8,

but if not you can always figure it out this way. 

This is an AoPS question and you’ve not demonstrated any solution method for your answer.   

Why the opposition to linking to other questions? Did the swill from jollyjellyjojo nauseate you?

According to him, the official answer from AoPS is (1/4).  With a name like jollyjellyjojo how could you go wrong?  

In these problems, simplify the radical so its easier to understand.

\(\phantom{=\quad}\sqrt{18\cdot n\cdot34}\\~\\ {=\quad}\sqrt{3\cdot3\cdot2\cdot n\cdot2\cdot17}\\~\\ {=\quad}\sqrt{3\cdot3\cdot2\cdot 2\cdot17\cdot n}\\~\\ {=\quad}\sqrt{3\cdot3}\cdot\sqrt{2\cdot 2}\cdot\sqrt{17\cdot n}\\~\\ {=\quad}3\cdot2\cdot\sqrt{17\cdot n}\\~\\ {=\quad}6\sqrt{17n}\)

There must be a pair of  17's  in the prime factorization of the number under the sqrt.

So  17  must be a factor of  n  , and the least possible positive integer value of  n  is  17

\(\phantom{=\quad}6\sqrt{17\cdot17}\\~\\ {=\quad}6\cdot17\\~\\ {=\quad}102\)_

There are  5  choices for the first letter, then 26 choices for the second letter, then 26 choices for the third letter.

so total number of possible words   =   5 * 26 * 26   =   3380

What interval consists of all \(w\) which satisfy \(\bf neither\) \(-2(6+2w)\le -16\) nor \(-3w\ge 18\) ?

If  w  makes the inequality   -2(6 + 2w) ≤ -16   false, then

it must make the inequality  -2(6 + 2w) > -16   true.

-2(6 + 2w)  >  -16

                             Divide both sides by  -2, a negative number, so flip the sign

6 + 2w  <  8

                             Subtract  6  from both sides of the inequality.

2w  <  2

                             Divide both sides by  2, a positive number, so don't flip the sign

w  <  1

If   -2(6 + 2w) > -16   then   w < 1

If  w  makes the inequality   -3w ≥ 18   false, then

it must make the inequality  -3w < 18   true.

-3w  <  18

                  Divide both sides by  3, a negative number, so flip the sign.

w  >  -6

If   -3w < 18   then   w > -6

So the solution to the question is all  w  such that   w < 1   and   w > -6

That is all  w  in the interval   (-6, 1)

expand    (6 x + 1/(3 x))^2.............................................(1)

[36 x^2 + 1/(9 x^2) + 4] * (6 x + 1/(3x))..........................(2)

[216 x^3 + 1/(27 x^3) + 36 x + 2/x] * (6 x + 1/(3x))..............(3)

[1296 x^4 + 1/(81 x^4) + 288 x^2 + 8/(9 x^2) + 24] * (6 x + 1/(3 x))........(4)

[7776 x^5 + 1/(243 x^5) + 2160 x^3 + 10/(27 x^3) + 240 x + 40/(3 x)] * (6 x + 1/(3 x))....(5)

46656 x^6 + 1/(729 x^6) + 15552 x^4 + 4/(27 x^4) + 2160 x^2 + 20/(3 x^2) + 160...............(6)

youre actually right.......!

Go online and  learn or modify Python code to generate nth fibonacci number here:

https://www.geeksforgeeks.org/python-program-for-program-for-fibonacci-numbers-2/

I'm seeing 1 valid arrangements using 1 blue {1}

5 valid arrangements using 2 blues {1x}, x=2-5

2 valid arrangements using 3 blues {124},{136}

and that's it

8 valid arrangements, 2⁶ = 64 possible arrangements

P[valid]=8/64 = 1/8

For these types of problems, just multiply by the conjugate like some people said, and you will get a+bi form.

This is very simple! Don't be afraid of imaginary numbers guys! If you want to simplify in the form a+bi, like \(\frac{2i+1}{i+3}\), multiply it by the conjugate of the denominator where it is also 1.

So for \(\frac{2i+1}{i+3}\), you multiply it by \(\frac{i-3}{i-3}\), which is the same as 1. This will simplify and won't change the fraction as you are basically multiplying it by 1. Remember to ALWAYS multiply by the conjugate.

This is your method for problems like these

I got \(\dfrac{1}{8}\), but I am not so sure either. Will wait for someone else to answer.

\(a - b \equiv n \pmod{99}\\ 62 - 75 \equiv n \pmod{99}\\ n\equiv -13 \pmod{99}\\ n \equiv 86 \pmod{99}\\ n \equiv 990 + 86 \pmod{99}\\ n \equiv 1076\pmod{99}\)

So n is 1076