geno3141

Since no one else has posted an answer, I'll post what I have ...

Using 'a' as the side opposite angle(A), 'b' as the side opposite angle(B), and 'c' as the side opposite angle(C):

Since angle(B) is a right angle:  sin(A) = a/b     cos(A) = c/b     tan(A) = a/c     b² = a² + c²

sin(A)  =  2·cos(A) + tan(A)     --->     a/b  =  2(c/b) + a/c

          multiplying by  bc          --->      ac  =  2c² + ab

Solving for  c  by using the quadratic formula:  ac  =  2c² + ab

                                                         2c² - ac + ab  =  0

                                c  =  [ a +/- sqrt( a² - 8ab ) ] /4

Since the answer is a real number:  a² - 8ab  >= 0

                                                                   a²  >=  8ab

Since  a  must be positive:                         a  >=  8b

which means that  a  must be larger than  b  but  b  is the hypotenuse!

So, either the problem, as stated, is impossible or I messed up!  Help!

Since O is the circumcenter, AO will be the circumradius (as will also be OB and OC).

Let X be the midpoint of AB.

Draw OX; OX will be perpendicular to AB.

Since AB = 10, AX = 5.

Triangle(AXO) will be a right triangle with right angle(AXO) and hypotenuse AO.

cos( angle(AX0) ) = 5 / AO

cos( 50^o ) = 5 / AO     --->     AO  = 5 / cos(50^o)

A right triangle with side lengths  6  and  b  has hypotenuse value  c.

Using the Pythagorean Theorem:  6² + b²  =  c².

The perimeter of the triangle is  24     --->     6 + b + c  =  24

                                                            --->                 b  =  18 - c

Substituting:                  6² + (18 - c)²  =  c²

                        36 + ( 324 - 36c + c² )  =  c²

                                     360 - 36c + c²  =  c²

                                             360 - 36c  =  0

                                                     -36c  =  -360

                                                           c  =  10

It's also undefined when a = 0,

Expanding  (2x + 5)(3x² + px + 6)  

                     =  6x³ + 2px² + 12x

                                +15x² + 5px + 30

                     =  6x³ + (2p + 15)² + (5p + 12)x + 30

Since the coefficient of the x term is half the coefficient of the x² term:

                      5p + 12  =  ½(2p + 15)

                  2(5p + 12)  =  2p + 15

                    10p + 24  =  2p + 15

                              8p  =  -9

                                p  =  -9/8

After you find the first root  1 + i

you can picture finding the other roots by making (around the origin)

a 90^o rotation, giving you  -1 + i

a 180^o rotation, giving you  -1 - i

a 270^o rotation, giving you  1 - i

A 360^o rotation will take you back to your original root.

There will be four fourth roots, separated by 90^o [ 360^o / 4  =  90^o ]

There will be three third roots, separated by 120^o

There will be five fifth roots, separated by 72^o

Etc.

I believe that one answer is  x = 1  y = 4  z = 5.

Another answer is  x = 1  y = 3  z = 8.

Another answer is 4 rows of 82 chairs.

Labeling the parallelogram in this fashion.  AB = 30  BC = 20  CD = 30  DA = 20  AC = 40.

One way to solve this problem is to use the Law of Cosines in this fashion:

1)  Find the size of angle(CAB) by using the Law of Cosines on triangle(CAB).

2)  Find the size of angle(DAC) by using the Law of Cosines on triangle(DAC).

3)  Add the results of steps 1) and 2) to find the size of angle(DAB).

4)  Find the length of side BD by using the Law of Cosines on triangle(DAB).

Let  a₁  be the first term.                                      Let  d  be the common difference.

Then:  a₂  =  a₁ + d         a₃  =  a₁ + 2d         a₄  =  a₁ + 3d         a₅  =  a₁ + 4d

a₅ / a₃  =  3     --->     ( a₁ + 4d ) / ( a₁ + 2d )  =  3

                        --->     a₁ + 4d  =  3( a₁ + 2d )

                        --->     a₁ + 4d  =  3a₁ + 6d

                        --->          -2a₁  =  2d

                        --->             a₁  =  -d

a₄ / a₂  =  ( a₁ + 3d ) / ( a₁ + d )  

Substituting  -d  for  a₁:    a₄ / a₂  =  ( -d + 3d ) / ( -d + d )  =  2d / 0     <---     Impossitble!