CPhill

5 , 6  and  x

By the triangle inequality, we have two situations

We have that

5 + 6  >  x

11 > x

So   x  < 11

And

5  + x  > 6

x  > 1

So...we have the following dual inequality

1   <  x <  11

So....the possible integer values for x are   2,3,4,5,6,7,8,9,10

And the sum of these  is  just   (10) (11)  / 2   - 1  =   55 - 1   = 54

The area  of the two  small squares combined  is 2 units^2

Note that   CY  = AY  -  AC  =  AY - AB   =  5  - 1   =  4

And  CD  = AB  = 1

So...the area of triangle  CDY  = (1/2) CY * CD  =  (1/2) (4) ( 1)  =  2 units^2

And, by symmetry, triangles   BDS , YEG   and SEF will have the same area as triangle CDY

So...the area of quadrilateral DYES  =

Area  of square  ASHY  less  areas of the two smaller squares  less areas  of the four triangles  =

5^2  -  2 -   4 (2)  =

25  - 2  - 8  =

25  - 10  =

15 units^2

The sum of the interior  angles  of the heptagon will be  (7 - 2) * 180   = 900°

So...each  angle  will measure   (900/7)°

And the sum  of the interior angles in the quadrilateral  is  (4 - 2)* 180  =  360°

So...in the quadrilateral  we  have that the sum of the angles  = 360°

x + x  +  2 (900/7)  =  360      simplify

2x +  1800/ 7   =  360

2x  =   360  - 1800/7

2x  = [ 7 * 360   - 1800 ] / 7

2x  =  [720] / 7      divide both sides by 2

x  =  [ 720 ] / 14

x  = [360 / 7 ]°

By the triangle inequality  we have that

S  + 33 > 42     where S is the shortest side

Subtract 33 from each side

S  >  9  units

So...since the shortest side has an integer length.....this length must be  10 units

So....the least perimeter  is    10  + 33 + 42   =   85 units

Note that  x^2  + 7x  - 44   can be factored as   ( x + 11) ( x - 4)

So   x^2 + 5x + a     could be  ( x + 11) ( x - 6)   and a  = -66

Or ..... x^2 + 5x + a  could be  ( x - 4) (x + 9)  and a  = -36

Note that in the first case   the  quotient is  ( x - 4)  /( x - 6)

And in the second case the quotient is  ( x + 11) / ( x + 9)

So...S  = ( -66, -36)   and the sum of its elements is  -102

We  have  C (10,3)  =  120  possible total selections of any three people

And  there  10  possibilities of selecting  3 people who sit together

( The first group occupies  seats 1, 2 and 3  and the last group occupies seats 10, 1 and 2)

So

1 2 3

2 3 4

3 4 5

.

.

.

10 1 2

So....the probability is just    10 / 120    =  1 /12

The interior angles of a pentagon will sum to 540°

So  we have that

A  = D - 40  

B  = D - 40

C  = D - 40

E  = D

So we have that

A + B + C + D + E  = 540      and substituting, we have

D - 40  + D - 40  + D - 40 + D  + D  = 540    simplify

5D - 120  = 540     add 120 to both sides

5D  =  660     divide both sides by 5

D  =  132°

Let the smallest term  = a

Let the  next term  = a + d

Let  the largest term  be  a + 2d

And we know that     

 (a + 2d)  -  a  = 14

2d  =  14

d  = 7

So  we have that

a  + a/2  +  (a + 7) + a/2  + (a + 14) + a/2  = 120  simplify

3a +  3a/2  + 21 =  120     

4.5a  + 21  = 120      subtract  21  from both sides

4.5 a  = 99    divide both sides by 4.5

a  = 22 

So...the smallest term is 22

Let  x  be the number of coins collected by  John

And  Mary  has three times  as many  = 3x

So

x  +  3x  =  120

4x  = 120       divide both sides by 4

x  = 30

So...Mary  has   3x  =  3 (30)   = 90 coins

You  are correct  !!!