CPhill

5 : 9 : 11

Call the number of  cookies  baked =  N

Alice baked  5 / ( 5 + 9 + 11) N =    (5/25)N  = (1/5)N

Susan baked  9/(5 + 9 + 11) =       (9/25)N

Terry baked     11/ ( 5 + 9 + 11)  =  (11/25)N

After each gave 12 cookies to  their  mother   Terry had   (11/25)N - 12  left   and this equals [ 4/ (1 +3 + 4)] ( N - 36)

So

(11/25)N - 12 =  [4/ (1 + 3 + 4)] (N - 36)

(11/25)N - 12 =  (4/8)(N - 36)

(11/25)N - 12 =  (1/2)(N - 36)

(11/25)N - 12 = (1/2)N - 18

(22/50)N - 12 = (25/50) - 18

(25/50) N - (22/50)N  = 18 - 12

(3/50)N  = 6

N = (50/3) * 6 =  100

Terry baked (11/25)(100) =  44

Susan baked (9/25)(100) = 36

Alice baked (1/5)(100) = 20

After each gave 12 to their  mother

Terry had  32

Susan had 24

Alice had 8

And      8 : 24:  : 32   =   1 : 3 :  4

Sam had 90 more than  Jim translates to :

S = J + 90         

He gives 140 to his  brother and 30 to Jim and  now  has (1/2) of what Jim has

S - 140 - 30  =  (1/2)J       

(J + 90)  - 140 - 30  = (1/2)J             {  remember  S =  J + 90 }

J -  80  = (1/2)J

J - (1/2)J = 80

(1/2)J = 80         

J = (2/1) * 80  =   160

S = 160 + 90  =  250  at first

Consider  a smaller set of   numbers   { 4 , 6, 8 , 10)

And we  wish to  choose any  2 of  these

The sums  are

4 6   =10

4 8  = 12

4 10  = 14

6 8 = 14

6 10 = 16

8 10  = 18

So.....we can get  every even integer  from 10  to 18 inclusive   =  (18 - 10) / 2 + 1 =  5 unique sums

So

The smallest sum in our list is

4  6  8  =  18     

And the largest is

12  14  16  =  42

We can get every  even integer from 18 to 42 inclusive

The number of unique sums =  (42 - 18) / 2  + 1 =   24 / 2 + 1 =  12 + 1   =  13  

The center is   (-5, 2)     

So we have that

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

x^2  + 10x + 25  + y^2 - 4y + 4   = r^2

x^2  + y^2 + 10x   - 4y   =  r^2   - 29      multipy through by 2

2x^2 + 2y^2  + 20x  - 8y  = 2r^2  -58

A = 2   B   = 20  C  = -8

And we  know that

2r^2  - 58  = 50       divide through by 2

r^2  - 29  = 25

r^2  = 54 

r =sqrt (54)

A + B + C  + r =   2 + 20  - 8  + sqrt (54)  =    14 + sqrt (54)  = 14 + sqrt (9 * 6)  = 14 + 3 sqrt (6)

P  and R  are  opposite angles in the quadrilateral  such that

P  = (3/5)R           so R > P

So

P  + R  = 180

(3/5)R + R  = 180

(8/5)R = 180

R = 180 * 5/8  =   112.5°

And Q =  (4/5)R = (4/5) (112.5)  =   90°

So  S  must also   = 90°   since Q and S are supplemental

So  R  is the largest angle

Let   x  be the cost of 1kg of sugar and y = cost of 1 kg of flour

We have these two equations

3x  +  (1/2)y  =   9.40

6x +     2y   =   20.20

Multiply the first equation by  -2  and add to  the  second

-6x - y  =  -18.80

6x  + 2y  = 20.20

______________

           y  =  1.40  =   the cost of  1 kg of  flour

Rearrange as

3x^2 -8x + c + 10

This will have two real roots when  the  discriminant >   0

So

(-8)^2  - 4 (3) (c + 10)  >  0

64 -120 > 12c

-56 > 12c

c <  -56 / 12

c <  -14/3

There are no positive integer  values for c that give two real roots !!!

Max  is   1 + 2/sqrt 3

Min is  1   - 2/sqrt (3)

Sum  = 2

We can let the center of the  diagram = (0, 0)  = O

 Let the very top circle (on which B lies) have the equation  x^2 + (y - 1)^2 =  1

The center of the  circle = E =  (0,1) and the radius =  1

Call the intersection of this  circle and  the next one in a clockwise direction  = C

So...the distance between AB and BC is the same

Triangle OBC  can be formed  and angle BOC  is an inscribed angle in the top circle with a measure of 360 / 12 = 30°

Then  the central angle in this circle (BEC)  has a measure of 2*30  = 60°

So triangle  BEC  has one angle of 60°

And the sides  EB and EC  are radii...so they are equal  = 1

And the measure of angles EBC and ECB opposite these sides   are  equal   =  (180 - 60)  / 2  =  60°

So triangle EBC is  equilateral because all angles  = 60°

So....in triangle EBC...EB = EC = BC =  1 in    =  AB

x^2-14x+3y+70=15+9y-y^2      simplify as

x^2 -14x + y^2 -6y  =  -55         complete the square on x, y

x^2 - 14x + 49  + y^2  - 6y + 9  =  -55 + 49 + 9

(x - 7)^2  + ( y - 3)^2  =  3

This is a  circle with a center of ( 7, 3)  and a  radius of sqrt 3

The total area is  formed by a  triangle  and  an arc in  the circle  of 360° - 131.81°  = 228.19°

The area of the  triangle  = (1/2) (sqrt 3)^2 * sin (131.81°) ≈  1.118

The area of the  arc is  pi (sqrt 3)^2  (228.19 / 360)  ≈ 5.974

The total area ≈  1.118 + 5.974 ≈  7.092