CPhill

Thanks.....I have a feeling that there is  some easy algorithm available to derive mods of negative integers, but I do not know what it is......

Connect RA  and  SA

Note that since  RA  = SA   (they are both radii)  then triangle RAS  is isosceles

So  angle  SRA = angle TRA=    [ 180  - angle RAS  ]  / 2 =    [ 180 -74 ]  / 2  =   90 - 37  =  53°

And angle RTB = angle RTA

And considering  triangle RAT,   by the exterior angle theorem,  angle    RAB =  angle TRA  +  angle RTA 

Angle RAB = 53  + 23  =  76°

But angle  RAB  is a central angle  equaling the measure of  minor arc BR  =  76°

-811   mod 24

One way of doing this  :

Thinking "backwards".....note  that  811/24  =   33.79......

Round this up to the  next integer =  34 

And note that   34 *24  = 816

The  residue  =   816 - 811   =   5

See the following  :

Note that this boils down to  finding  the area  of  the quarter circle  with a center at the origin and with a radius of 1 within a square with a side of 2

Area of the quarter circle  = pi (1)^2 / 4   =   pi/4

Area of the square  = 2^2   = 4

Probability =    area of quarter circle  / area of square  =   (pi/4)  / 4  =   pi / 16 

We can use something known as Heron's Formula to solve this

First.....find the semi-perimeter, S   =   [ 13  + 17  + 12sqrt (3)  ] / 2  =  15 + 6sqrt (3)

Area   =   sqrt  [  S  * ( S -A)  *  (S - B)  * ( S - C) ]  .....  [  A,B, C  are the triangle sides ]

Area  = sqrt  [  (15 + 6sqrt (3) )   *   ( 15 + 6sqrt (3)  - 13) *  ( 15 + 6sqrt(3) - 17)  *  (15 + 6sqrt (3) - 12sqrt(3)) ]  =

sqrt  [ (15 + 6sqrt (3))  * (2 + 6sqrt(3)) * (6sqrt (3) - 2)  *  (15 - 6sqrt (3)) ] =

sqrt  [ ( 6sqrt(3) + 2) * ( 6sqrt (3) - 2) * ( 15 + 6sqrt (3)) * ( 15 -6sqrt (3) ) ]  =

sqrt [  (36*3 - 4)  * (225- 36*3) ]  =

sqrt  [ 108 - 4) * ( 225 - 108) ] =

sqrt  [ 104  * 117]  =

sqrt  [ 26 * 4  * 13 * 9] =

sqrt [ 13 * 2  * 4  * 13  * 9 ]  =

sqrt [ 13^2  * 2^2 * 3*2  * 2  ] =

13 * 2 * 3  * sqrt (2)  =

78 sqrt (2)  units^2

We can choose any of the 5 red shirts

Any of the 4 pairs of pants

Any of the (7 green  + 8 red) hats  = 15   (we canot choose  the same  color of red as the shirt we chose)

So...the total number of outfits  =   5 * 4 * 15  =  300

Thx, hugoslav  !!!!

Melody  has set up a partial fraction problem......thanks, Melody!!!

It may not be entirely clear  what to do after her last step

Note  that if we multiply through by the common denominator of  (x-3) (c-5)  we get that

2x+ 7 = A(x-5)  + B (x - 3)

2x + 7 = (A + B)x  -5A  - 3B 

This implies  that we have this system  :

A + B =  2

-5A - 3B = 7

Manipulating the first  we have that  B  = 2  - A

So....we  have  that

-5A - 3 ( 2 -A)  = 7

-5A  -6 + 3A  = 7

-2A  =  13

A = -13/2

And B  = 2 -  - (13/2)  =  17/2

Well....of course......the combined area of the  six small circles is just  6 * pi (radius)^2  = 6 pi (3)^2  = 54pi

Note that  if   we  connect  the centers  of  the top two circles  and we  connect the center of the  large circle with both of these centers,, we  will  have an isosceles triangle  with one side of 6 , two sides of x  and an included angle  between these unknown sides of 60°

Using the Law of Cosines, we can find  x  thusly......

6^2  = 2x^2  -  2x^2  (cos 60)

36  = 2x^2 - 2x^2 (1/2)

36  = x^2

x  = 6

So.....the radius of the larger circle  = x + 3  =   6 + 3   =   9

So  the area  inside rhe larger circle  but  outside the smaller circles = 

pi  [ 9^2  - 54]  =

pi [ 81 - 54 ]  =

27 pi  units^2

Call the point  where  BT  intersects the circle, A

And  we have  that

Measure of angle RTB  =  (1/2)  ( measure of minor arc RB - measure of minor arc SA)

But angle SOT  = measure of angle  SOA

And since SOA  is a  central angle, it has  the same measure as minor arc SA

And.....since ROB is also a central angle, it has the same measure as minor arc RB

Call the meaure of  minor arc SA =  angle SOT  = angle SOA  =  M

And call the measure of minor arc RB  = angle ROB =  4M

So  we have this equation

28  = (1/2) (4M - M)

56  = 3M

56/3  =  M

So  ,minor arc  SA = (56/3)°

And minor  arc RB   =  4 (56/3)  = (224/3)°

And  minor arc RS  =   180  - 56/3 - 224/3   = (260/3)° ≈  86.66°