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We have that

XY/ sin Z  = XZ / sin Y

XY/ sin 60°  = 4 /  sin 45°

XY  =  4sin 60°  /sin 45°

XY  =  4 sqrt (3)  / sqrt (2)

XY =  4 sqrt (6)  / 2  =   2sqrt (6)

By the Sine Law, XY = 3 - sqrt(3).

Thank you so much!!!!!!!

2^9.226412193 = 599        2⁰ = 1         9.226412193 + 0 = 9.226412193

When x = 0   you find   c = 5

x^2 + bx + 5       now, when x = 1

1 + b + 5 = 10      so b = 4

x^2 + 4x + 5        

       Hmmmmm    something seems amiss     when x = 2  y should be 17 , not 19   

                                 did you enter your Q correctly?

A cubic yard is    3 ft x 3 ft x 3ft = 27 ft ³

12 bags @ 2 ft³  = 24 ft ³

24 / 27 = 8/9 of a cubic yard

DX / DY = DY / DZ

DX = DY² / DZ        Everything else is easy.

THANKS SO MUCH!!!

IDK OK

[Q + Q + 4 + Q - 1 +3Q + 12]=27, solve for Q

Q = 2 - Third pile

2 + 4 = 6 - Second pile

2 - 1 = 1 - First pile

6 * 3 = 18 - Forth pile

[2 + 6 + 1 + 18]==27 Quarters.

x + 1/2 x + 1/4 x + 1/8x + 1/16x = 310

31/16  x = 310

x = first day = 160 pages

Hmmmm....WHY is this posted THREE times !????

Second pile = x

first pile = x-5

third pile = x -4

fourth pile = 3x     totalled = 27

x + x-5 + x-4 + 3x = 27

6x =36

x = 6        fourth pile = 3x = 18 

a=1, b = -2, c=2
A few examples:
20 = (1, 2, 4, 5, 10, 20) > 6 (divisors)
20^[a6 + (-2)] / 2 =2 =20^2 =400=[10*5*4*2*1]

64 = (1, 2, 4, 8, 16, 32, 64) > 7 (divisors)
64^[a7 + (-2)] / 2 =2.5=64^2.5=32,768=[32*16*8*4*2*1]

24 = (1, 2, 3, 4, 6, 8, 12, 24) > 8 (divisors)
24^[a8 + (-2)] / 2 =3 =24^3=[12*8*6*4*3*2*1]

36 = (1, 2, 3, 4, 6, 9, 12, 18, 36) > 9 (divisors)
36=^[a9 + (-2)] / 2 =3.5=36^3.5=[18*12*9*6*4*3*2*1]

48 = (1, 2, 3, 4, 6, 8, 12, 16, 24, 48) > 10 (divisors)
48^[a10 + (-2)] / 2 =4 =48^4=[24*16*12*8*6*4*3*2*1]

60 = (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60) > 12 (divisors)
60^[a12 + (-2)] / 2 =5=60^5=[30*20*15*12*10*6*5*4*3*2*1]

So: a + b + c =1 + (-2) + 2 = 1

The perfect squares from 1 through 2500 inclusive, are printed in a sequence of digits 1491625.....2500  What is the 100th digit in this sequence?    

By brute force I found it to be 9,  You'd better check my figures though just to be sure. 

                  Squares                                      # in row      Running Total 

1 4 9 16 25 36 49 64 81 100                            18                   18  

121 144 169 196 225 256 289                         21                   39  

324 361 400 441 484 529 576 625                  24                   63  

676 729 784 841 900 961 1024 1089              26                   89  

1156 1225 1296                                               12                 101  

1491625364964811001211441691962252562893243614004414845295766256767297848419009611024108911561225129

The 100th digit is 9, which comes from 36^2==1296

Just list them out and count to 100th one

1491625 ... 36 49 64 81 100 121 144 169 196 225 256 289 325 361 400 441 484 529 576 625 676 729 784 841 300 961 1024 1089 1156 1225 1296

         1369 1444 1521 1600       

They each got 1/3   of the remaining 6/7

6 /7   * 1/3 = 6 /21 of a pie

There is six ways to arrange the three

a b c

c a b

b c a

a c b

c b a

b a c               

      then there are 6 ways to seat them next to each other in the row of 8 chairs

6 x 6                

   and there are 5 remaining people to arrange in the remaining 5 seats

                                5!

5! x 6 x 6 = ...............

Remember that |a + bi| = a^2 + b^2.

So from equations, (z - 3)^2 + (z - 5)^2 = 4 and (z - 6)^2 + (z - 6)^2 = 4.

(x - 3)^2 = 4, (y - 5)^2 = 4, (x - 6)^2 = 4, (y - 6)^2 = 4

Two possible values of x, two possible value of y.  Therefore, four solutions for z.

The sum of the areas of the black regions is 2pi square units because the sum of interior angles of a hexagon is 2 full circles or 720 degrees.

Thx, Pangolin  !!!!

We can construct a rifght  triangle with legs of  (5 - r)   and a hyotenuse of  (5 + r)  where r is the radius of the smaller circle

So by the Pythagorean Theorem  we have

2 ( 5 - r)^2  =  (5 + r)^2     simplify

2 ( r^2 - 10r + 25)  =  r^2  + 10r   + 25

2r^2 - 20r + 50   =r^2  + 10r   + 25

r^2 - 30r  +  25  =  0       complete the square on r

r^2 - 30r  + 225  =  -25 + 225

(r - 15)^2   = 200       take the smaller root

r  - 15  =  -sqrt (200)

r =   15  - 10sqrt (2)

So  the blue area   =    (area of quarter square    -area of  quarter circle  -area of small circle ) =

10^2/4  - (25/4)pi   - (15 - 10sqrt (2))^2*pi   ≈  3.05 units^2

Let the fractions be x and y. Then, we have that x + y = 11/12 and xy = 1/6. We also have that y = 11/12-x, so x(11/12-x) = 1/6. Thus, 6x(11/12-x) = 1, and 66x/12 - 6x^2 = 1. Simplifying, 11x/2 - 6x^2 = 1, and we have a quadratic that we can solve with the quadratic formula, getting x = 1/4 or 2/3. Thus, the lesser fraction is 1/4.

By the product formula, the product of the proper divisors of n is n^((2n + 1)/2), so a + b + c = 2 + 1 + 2 = 5.