CPhill

I actually like your way better, X²....!!!!

Note that the sum   of

2 + 4 + 6 +....+ N         can  be written as  (N + 2) * (N /4) 

And the sum of  

1 + 3  + 5 +  ....+ N     can be written as  [ (N + 1)/2]^2  =  (N + 1)^2 / 4  =

So  we have 

[ 2009 + 1 ] ^2 / 4      -  [2008 + 2] [2008] / 4 =

[2010]^2 / 4   -   [ 2010] *[ 2008] / 4  =

[2010] / 4  * [ 2010  - 2008]  =

[ 2010 ]/ 4  *  2  =

1005

Here's one more approach....

Note that the points    (0, - 2)  (3,1)  and (4,0)  are on the graph

So....we must have that

0 =  4 + A    ⇒  A  =  -4

And

-2  =  [0 + (-4)]  / [B (0) + C ]

-2  +  -4 / C

-2C  = - 4    ⇒  C  =  2

And

1  =  [ 3 + (-4) ] / [ B(3) + 2 ]

1 =  - 1 / [ 3B + 2]

3B + 2  =  - 1

3B =  -3  ⇒   B =  -1

So....A + B + C     =     -4 - 1 + 2    =    -3

The center of this circle will lie on the remaining vertex.....

The  intersection of these two areas is just 1/4 of the circle's area  =

pi* r^2  / 4  =     pi (3)^2 / 4    =    (9/4) pi  units^2

6)An obtuse triangle has integer length sides. If two sides of the triangle are 16 and 21, how many possible lengths are there for the third side?

Let  M be the missing side

For  M to be a possible solution, we must have that

M^2  > 16^2 + 21^2  

M^2 > 697      

M > ≈ 26.4

So  the shortest integer length for M  is 27

But...

21 + 16 > M   which means that  M < 37

So...the longest integer length for M   is 36

So...there are  36 - 27 + 1   = 10 possible side lengths

See my corrected answer.....

5)A triangle has integer length sides. If two sides of the triangle are 16 and 21, how many possible lengths are there for the third side?

We have to have that

If the missing side is the shortest side  then

Shortest side + 16 > 21

So...the shortest side > 5

And the longest side must be

16 + 21 > longest side

Longest side < 37

So....the possible sides are

5 < possible sides < 37               }

So the possible sides are   36 - 6 + 1     =   31 possible lengths

4)The distance from Capital City to Little Village is 660 miles. From Capital City to Mytown is 310 miles, from Mytown to Yourtown is 200 miles, and from Yourtown to Little Village is 150 miles. How far is it from Mytown to Little Village?

Here's the layout [  assuning that all the cities are collinear ]

C                  660                                              L

C       310             M

                            M     200       Y 

                                                  Y      150       L

Mytown to Little Village  =  200 + 150    =  350 miles

3)A triangle with integer sides has perimeter 12. How many such non-congruent triangles are there? (A 3-4-5 triangle is considered congruent to a 3-5-4 triangle because we can reflect and rotate the triangles until they match up.)

Possibilities :

3 - 4 - 5

4 - 4 - 4

5 - 5 - 2