CPhill

Before Grace starts rowing Will has rowed   45 * 50  = 2250 m

So  the distance between them before Grace starts rowing is  2800 - 2250  = 550 m

When Grace starts rowing, the number of  minutes until they meet is

550 / (50 + 30) =  550 / 80 =  55/ 8   =  6.875  = 6 + .875 (60)   = 6 min 52.5 sec

So....they meet at    2:45  + 6 min 52.5 sec =  2:51 and 52.5 sec

Let M be the midpoint of  BC

BM =  1

Let O be the center of the red circle  and r  its radius

OB =  (radius of large  circle - r)  = (2 - r)

OM = (1 + r)

Triangle OBM is  right with OM the hypotenuse

So

OB^2 + BM^2  = OM^2

(2 - r)^2 + 1^2  = (1 + r)^2

r^2 - 4r + 4 + 1  =  r^2 + 2r + 1

6r = 4

r = 4 / 6  =  2 / 3

Here's my best attempt

4 A's

Choose  any 4 of 6 positions  for the A's  = C (6,4)

For the other two positions we can  choose 1 of 25 remaining letters for one position  and then 1 of any 24 remaining letters  (We can't choose  2  B's )

So 

C(6,4) * 25 * 24  =  9000

2 B's  

Choose any 2 of 6 positions for the B's

For the other positions  we can  choose any 1 of  25 letters for 1 position and (24)^3 for the other 3 positions ( we can't have 2 A's)

C(6,2) * 25 * 24^3 =  5,184,000

4A;s, 2B's  = All the arrangements for the "word" AAAABB  =    6!  / ( 4! * 2!)  =  15

I get

9000 + 5,184,000 + 9000  = 5,202,000  plates

 Let AD be the altitude  and BC the base

tan B =  AD / BD  =  3/5  =    3 / 5

tan C = AD / CD  = 1/4 =       3 / 12

BD  =  5 / (12 + 5) *32  =   (5/17) * 32  =  160 / 17

AD / BD =  AD / (160/17)   = 3/5

AD =    (160 / 17) *( 3 / 5) =  96 / 17

[ABC]  = (1/2) BC * AD  = (1/2) 32 * 96 / 17  =  1536 / 17

x^3 + x^2 -3x + 5 =  x^3 + 2x^2

2x^2 + 3x - 5  = 0

(2x + 5) ( x -1) = 0

2x + 5  = 0                                                   x - 1 =  0

x = -5/2                                                           x = 1

y = (-5/2)^3 + 2(-5/2)^2 = -25/8                y  = (1)^3  + 2(1)^2  =  3

(-5/2, -25/8) ,  ( 1, 3)

If all the Psychology textbooks are together then so are all the Math textbooks

Arrange the math textbooks in 4! ways  and the Psychology textbooks in 3! ways

And either the Math textbooks or the Psychology  can come first

2 * 4! * 3!  =    288 ways

                  A

          3               5

B               D                    C

                  7

Law of Cosines

7^2  = 3^2 + 5^2  - 2 (3 * 5) cos (BAC)

cos (BAC )  =  [ 7^2 - 3^2 - 5^2 ] / [ -2 * 3 * 5 ]  =  - 1/2 

BAC  = 120°

Since BAD is  a bisector

BAD = 60°

cos (BAD)  = 1 / 2

sin A =   BD / AB  =  BD / 24  = 1/3  →  BD = 8

sin C = BD / BC =  8 / BC = 1 / 2  →   BC  = 16

(x -1) (x + 1) =  x^2 -1  =  x^2 + 0x + (-1)

Simplify as

6x^2 - 21x -31 = 0

x =   [ 21 + sqrt [ 21^2 + 4(6)(31) ] ] / 12  =   [21 + sqrt (1185) ] / 12