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Thanks Heureka and Tiggsy.   

Suppose 1/x = 2/(y + z) = 3/(z + x) = (x^2 - y - z)/(x + y + z).  What is (z - y)/x?

Using the first two equalities  we have that

(y  + z)  =  2x       (1)

And using the second two equalities we have that

2(z + x)  = 3(y + z)  ⇒  2(z + x)  =  3(2x) ⇒   2z = 4x   ⇒  z = 2x   (2)

Sub (1) into the last expression in the equality :

(x^2 - (y + z) )                 x^2 - 2x        x ( x - 2)          x - 2

____________  =        ________   =  _______  =    ____

(x + y + z )                       x + 2x            3x                   3

So  using this and the first expression we have that

x - 2               1

____  =       ____      cross-multiply

   3                 x

x^2 - 2x = 3

x^2 - 2x - 3  =  0     factor

(x - 3) ( x + 1)  =   0

So either x = 3   or    x  = -1

Which means that   z = 6    or  z  = -2

Let x  = -1    and z =  -2

(y + z) = 2x

( y - 2)  = -2

Which means that y  = 0

Note that....when x = -1 , z =  -2    we have that

1/x = 2/(y + z) = 3/(z + x) = (x^2 - y - z)/(x + y + z)

1/ -1  = 2 / (0 - 2)   =  3 / (-2 - 1)   =  [ (-1)^2 - 0 + 2 ] / [ -1 - 0 - 2]  =  

1/-1   =  2/-2  = 3/ -3  =  3/-3

-1  = - 1  = - 1  = - 1

So

(z - y )  / x   =

(-2 - 0) / -1   =

2

BTW......The same result would be found when x = 3  and z  = 6

Let  the total watermelons we start with  =  N

1st  hour sold  =  [   N/2 + 1/2]

Remaining  is  what we started with - number sold in 1st hour =   N - [ N/2 + 1/2 ]  =  (1/2)N -1/2

2nd hour sold =   (1/2)[ (1/2)N - 1/2] + 1/2 =  [ (1/4)N  + 1/4 ]

Remaining is   what we started with at the end of the first hour - what we sold in the second hour  =

[ (1/2)N - 1/2 ] -  [ (1/4)N + 1/4]  =  [ (1/4)N - 3/4 ]

In the third hour  we sold  

(1/2)  [ (1/4)N - 3/4 ]+ 1/2    =   [ (1/8)N + 1/8]

What we  have remaining is the what we started with at the end of the second hour - what we sold in the third hour  =[(1/4)N - 3/4]  - [ (1/8)N + (1/8) ] =   (1/8)N - 7/8

And the number remaining  = 6  ...so...

(1/8)N - 7/8  = 6      multiplty through by  8

N - 7  =  48

N  = 55   is what we started with

Thanks! I would give you credit but... I would be failed for cheating. But this helped a lot, thanks!

To simplify things........call the third integer 2N....so we have that

(2N-4) + (2N -2)) + (2N) + (2N + 2) + (2N + 4) + (2N + 6)   = 282

2N (6)  + 6  =  282      subtract 6 from both sides

12N  =  276   divide both side by 12

N =  23

So....the integers are

2(23) - 4  = 42   and then   44, 46, 48 , 50 , 52

3 :  6  :  4

Call Natnan's share   N

Alesha gets N + 18 

And Jordan gets  ( 4/3)N

And we know that

(N + 18) / [(4/3)N]  =  6 / 4          cross-multiply

4 ( N + 18)  = 6 (4/3)N

4N + 72  =  8N

72  =  4N

18  = N

So   Jordan gets   (4/3)N  = (4/3)(18)  =  24  pounds

2.    Using the Law of Cosines

6^2 = 5^2  + 5^2  - 2 (5)(5) cos BAC

36 = 50 - 50 cos BAC

[36 - 50]

_______     = cos BAC

     -50

14

___  =  cos BAC

50

 7

__  =  cos BAC

25

So  sin BAC  =  24 / 25

And angle  BOC  = twice angle BAC

So

cos (2 * BAC)  = cos (BOC) =   1 - 2sin^2 (BAC)

cos (BOC)  = 1 - 2 (24/25)^2  =   1 - 2 ( 576/625)  =  [ 625 - 2(576) ] / 625  = -527/625

And using the Law of Cosines again

6^2  =  r^2 + r^2  - 2(r)(r) cos (BOC)

36 = 2r^2 - 2r^2 [ -527 / 625 ]

36 = 2r^2  [ 1 + 527/625 ]

18  = r^2 [ 1 + 527/625 ]

18 = r^2  [1152 /625 ]

r^2 =  [625 * 18 ] / 1152

r^2  = 11250 / 1152

r^2 =  625 / 64

r = 25/8  = the circumradius

So   the semiperimeter of triangle   BOC  is   [ 25/8+ 25/8 + 6 ]/2   =  49/8

So...using Heron's Formula....the area of BOC is

sqrt  [ (49/8) ( 49/8 - 25/8) (49/8 - 25/8) (49/8 - 6) ]  =

sqrt [ (49/8) ( 24/8) (24/8) ( 1/8) ] =

(7 * 24 * 1)              168              21      

__________  =        ___    =     ___  =   2.625 units^2

   64                          64               8 

N + 2N + 4/3N =(3 1/3)N+18, solve for N
N = 18 - pounds - Nathan's share
N + 2N + 4/3N = 78 - pounds in total
18 + 18 = 36 - pounds- Alisha's share
78 - 18 - 36 = 24 - pounds- Jordan's share.

2.

(x^2-4x+3)(x+5) - (x^2+4x-5)(x-c) = 0    simplify

(x^3 - 4x^2 + 3x) + (5x^2 - 20x + 15)  - (x^3 + 4x^2 - 5x - cx^2 - 4cx + 5c)

(x^2 - 17x + 15)  - 4x^2 + 5x + cx^2 + 4cx - 5c  = 0

(c - 3)x^2  + (4c - 12) x + (15 - 5c)  =   0

(c - 3)x^2 + 4(c - 3)x - 5(c - 3)  =  0

(c - 3) (x^2 + 4 - 5)  =  0

c = 3       will make this true

$6,500 - $2,000 = $4,500 to be deposited monthly

$4,500 / 36 months = $125.00 that must be deposited by you AND your grandparents.

$125 - $5 = $120 - monthly deposits that your GPs must make every month for 36 months.

1.

f(1)  =   3 - 7 + 2 - b + 1  =  1

So

-1 - b  = 1     add 1 to both sides

-b = 2

b = -2

Let the positions be   A  B   C   D     E     from left to right

There  are    C(5,2)  = 10 possible positions that the  boys can occupy

But  only 4 of them are together

AB  BC  CD   DE

So....6 of them aren't

And for each of these, the boys can be arranged 2 ways

And for each arrangement of the boys, the girls can be arranged in 3!  =  6 ways

So......the number of possible line-ups  =

6 * 2 * 6   =     72 possible line-ups

There are 24 ways to form "words"  when C appears first

And 24 ways to  form words whenever i  appears first

And 24 ways to form words when K comes first

And 24 ways to form words when M appears first

So  when U appears first, it will be in the 4 * 24 + 1 = 97th position

So we wiill have

UCIKM         UICKM    UKCIM    

UCIMK         UICMK    UKCMI

UCKIM         UIKCM    UKICM

UCKMI         UIKMC    UKIMC

UCMIK         UIMCK

UCMKI         UIMKC

So....it occupies  the 112th position

2) A radio telescope has a parabolic dish. Radio signals are collected at the focal point (focus) of the parabola. The distance from the vertex of the parabolic dish to the focus is 20 feet. The vertex of the dish is located at a point 30 feet above the ground and 80 feet to the east of a computer that reads and records data from the telescope. The diameter of the dish is 120 feet.

What is the depth of the parabolic dish?

We can let the parabola open upward......so we will  have the form

4a ( y - k)  =  (x - h)^2

The distance from the vertex to the focus =  20   = a

We can let the vertex be   ( 80, 30)

So we have the form

4(20) ( y- 30)  = (x - 80)^2   simplify

80 ( y - 30)  = ( x - 80)^2

Since the diameter  is 120 ft...the radius is 60 ft.....so we can let one point on the parabola be   ( 80 + 60 , a) =

(140 , a)......where a  is the height of the dish....so we have that

80 ( a - 30) = (140 - 80)^2

80 ( a - 30)  = (60)^2

80 ( a - 30) = 3600      divide both sides by  80

a -  30 =  45      add 30 to both sides

a = 75 ft  = the height of the dish

Here's the answer to the first question :

https://web2.0calc.com/questions/loran-question

a b = 2 (a + b)...............(1)
a b = 6 (a - b)................(2)

2[a +b] =6[a - b]
2a +2b =6a - 6b
a = 2b   Sub into equation (1)
2b^2 - 6b = 
2b(b - 3) = 0
b(b - 3) =0
b = 3    and     a=6

1) The side length of the original equiateral triangle is 12, so the area is decreased by 20*sqrt(3) cm^2.

2) The circumradius is 4, so the area of OBC is 3*sqrt(7).

x+y=9

The vertex is at (3,-5).
The parabola is opened downwards because there is a minus in front of the bracket.

oh..

Here's a different method, Newton Interpolation, sort of.

Let

 \(\displaystyle f(x) = a_{0}+a_{1}(x-1)+a_{2}(x-1)(x-2)+a_{3}(x-1)(x-2)(x-3)+(x-1)(x-2)(x-3)(x-4)\)

(The coefficient for the final term will be 1 since the polynomial is monic).

Sustituting x = 1, 2, 3 ,4, gets us

\(\displaystyle f(1)=1=a_{0}, \qquad\text{ so }\quad a_{0}=1,\\ f(2)=2=a_{0}+a_{1},\qquad \text{ so }\quad a_{1}=1,\\ f(3)=3=a_{0}+2a_{1}+2a_{2}\qquad \text{ so }\quad a_{2}=0,\\ f(4)=4=a_{0}+3a_{1}+6a_{2}+6a_{3} \qquad\text{ so }\quad a_{3}=0.\)

So,

\(\displaystyle f(x)=1+(x-1)+(x-1)(x-2)(x-3)(x-4)\\ \text{and therefore}\\ f(6)=1 + 5+5*4*3*2=126.\)

I'm getting N = 24.

p can be 2, 3, 11, and 19, so there are four possible values of p.

The superdeficient numbers are 8, 12, and 15, for a total of 3 superdeficient positive integers.