CPhill

(x + 1)  ( x -1)  =  0

x^2  +  0x - 1  = 0

  A

      4            5

     B        D          C

                3

Since AD is a  bisector.....Let BD = x  and DC  =3 - x   and  we have that

BD/ AB  = DC / AC

x / 4 = (3 - x) / 5

5x = 4 (3 - x)

5x  = 12 - 4x

9x= 12

x =  4/3  = BD

[ ABD ] = (1/2) (4/3)(4)  =  16/6 = 8/3

[ABC ]  = (1/2)(4)(3)  = 6

[ADC ]   =  [ ABC ] -  [ ABD ]  = 6 - 8/3  =   10/3 =  3 + 1/3  =   3 (to the nearest integer )

200 =  2^3 * 5^2 * 1   =  2*2*2*5*5*1

No. of distinct  sequences  =

6! / [ No. of times 2 appears ! * No. of times 5 appears ! ]

6! / (3! * 2!)  =   60

By  " brute force " .....

Laverne  says     2   7   12    17      22       27      32

Progressive sums     9      21    38        60       87      119

Another (more mathematical )  way

First term Laverne says = 2

Let the last term she says = 2 + 5n

Sum  > 100

[ 2 + (2 + 5n) ]  * n  / 2   > 100

[ 5n + 4 ] * n  >  200

5n^2 + 4n - 200 > 0      (1)            set =  0   and solve for positive n

5n^2 + 4n - 200 = 0

n = [ -4 + sqrt [ 16 + 4000] ] / 10    ≈  5.9

≈ 5.9 is a root of 5n^2 + 4n - 200....so....the next largest integer also  makes (1)  true

n = 6

And the number that Laverne says  is  2 + 5(6)  =    32

Not great at "counting" probs....here's my best attempt....corrections welcome !!!

1 3      5 6 7 8 9 10 11                7

2 4      6 7 8 9 10 11 12              7

3 5      7 8 9 10 11 12                 6

4 6      8 9 10 11 12 1                 6

5 7      9 10 11 12 1  2                6

6 8     10 11 12 1 2  3                 6

7 9     11 12 1  2  3  4                 6

8 10    12 1 2 3 4 5                     6

9 11     1 2 3 4 5 6                      6

10 12   2 3 4 5 6 7                      6

11  1    4 5 6 7 8                         5

12  2    5 6 7 8 9                         5

2(5) + 8(6) + 2(7)  =  72 ways

  See here

There are no students  only  in the Math club

3 Students are in  both  - 3 who are in the Science club are also in the Math club

There are 3 students  only in the  Science club

tan (7pi/4)

Note that this  really     tan ( 2pi - pi/4)  

So...it's just  a 45°  angle in the 4th quadrant =   tan (- pi / 4)

tan (pi/4)  =  1

But in the 4th quadrant the tangent  is  -

So

tan (7pi/4)  = tan (- pi /4)  =   -1

OPO  = (3/5)(2/4)(2/3)  = 1/5

POP  = (2/5)(3/4)(1/3)  =  1/10

1/5 + 1/10  =   3 / 10

1 step  =  4 points

2 steps  = 9 points

3 steps = 16 points

n steps  = (n + 1)^2  points

12 steps  = (12 + 1)^2  =  169 points

If you drove at 50 mph, you would have arrived  EARLIER, not, LATER