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Thanks , hugo.....here's  another  way

The  sum of the first 200 positive integers is  just  (200)(201)  / 2  =  100 * 201   =  20100

But we have actually  "over-summed"  the given series by

2 [ 4  +  8   + 12   + 16  +.....+  200 ]

The  number  of  terms  in the brackets  =  [ 200 - 4 ] / 4  +  1  =  196/4 + 1  =  50  

So  the sum of this series is  just

2  [  200 + 4 ] * (50/2]   =  204 * 50  =  10200

So.....the sum of this series =   20100  - 10200  =    9900

(6 - 4) + (18 - 8) + (30 - 12) + (42 - 16)+............+ (594 - 200) = sum_(n=1)^50(8 n - 6) = 9900

Let's  just get  the square of the distance formula

Note 10sqrt (2)  =  sqrt (200)

( 4 - x)^2  + (21 - 7)^2  =  (sqrt 200)^2         expand  and simplify

(x - 4)^2  + 14^2   = 200

(x - 4)^2  + 196  = 200

(x - 4)^2  =   4        take  both roots

x - 4  =  sqrt (4)        and     x - 4  =  -sqrt (4)

x - 4  =  2                            x - 4  =  -2

x  = 6                                  x  =  2

So...the sum    = 6 + 2    =  8

No prob, ava    !!!!

First....find  the  slope of the line  =  [ -2 -7] / [3 - -1] = -9/4

Next  we  can write  the equation of the line  using either point

y = (-9/4) ( x - 3) - 2

And  when  x  =  - 6

y = (-9/4) (- 6 - 3) - 2

y = (-9/4) (-9) - 2

y = 81/4  - 2

y = 81/4  - 8/4

y = 73/4

Thank you!

(x - 4)^2  + (y + 1)^2  = 25

The center of this circle  is   ( 4, -1)

We  can use the negative  reciprocal slope to  -3/4 =4/3  to  find the  points  where the tangent lines touch the  circle 

If we move 3 units to the right on x  and 4  units up on y from the center we will have the first point where a tangent line touches  this  circle    = ( 7, 3)

Likewise....if  we  move  3 units to the left and 4 units down from the center we will have the second point where a tangent line touches the circle  = ( 1, -5)

The equation of the  first line is

y = (-3/4)(x - 7) + 3

The equation of the second line is   

y =(-3/4)(x - 1)  -5

See the graph here  :  https://www.desmos.com/calculator/mztyclw9nw

((5 chose 2)+ (7 chose 2) + (4 chose 2))/((5+7+3+4) chose 2)

Not too difficult....

Just sub  4  into each x  in the equations  and see which one is true

Note  that  C appears to be correct  !!!

There is only one pair (a,b) that works, which is (4,2), so there is only one solution.

a. y = 5/4.

b. The x-intercept is (14/9,0).

Try plotting the two equations on a graphing calculator, like Desmos.com. The x-value of the intersection will be your answer!

It said it was wrong.

Sporcle is another site that many of you would enjoy.

It is a huge site with all kinds of things you can play with and learn.

I think you can write your own games too if you have some programming skills.

But among its many attractions it has some really good logic puzzles.

I have not played with them for ages but I think they are mostly here.

Thanks Hugo,

It is already in sticky Topics under

 @ Site full of printable puzzles.  

Since it's isoceles right, the angles must be 45-45-90.

That means that the hypothenuse is sqrt(200).

We just need to find the length of half the hypothenuse, since the unshaded is 45/360 of a circle.

sqrt(200)= sqrt(4*50) = 2(sqrt(50)).

The radius is (2(sqrt(50)))/2 = sqrt(50).

area of circle is π r^2, so it's 50π. The part is 45/360 of a circle, with two parts, so 90/360 = 45/180 = 1/4

50π/4 is the unshaded.

The triangle has an area of 10*10/2 = 50.

50 - 50π/4  is the area of the red.

If you don't understand anything feel free to ask!

Put this in sticky topics!

Thanks for sharing this list, appreciate it!

Intuitively...

You select any number, and there are, from 3 to 97,always 4 numbers satisfying this, so 4/100 = 1/25 probability that you get a difference of 1 or 2.

The probability of getting 3-98 is 96/100 = 48/50 = 24/25. Multiply together and get 24/625.

If you pick 2 or 99, with 2/100 = 1/50 probability, you get 3 numbers satisfying, so 3/100. Multiply together to get 3/5000.

If you pick 1 or 100, with 2/100 = 1/50 probability, you get 2 numbers, so 2/100 = 1/50. Multiply together = 1/2500

I'll let you add them together.

If you don't understand anythign feel free to ask.