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Remainder is 3

The area of the circle is 176 pi.

2-n/n+1 + 2n-4/2-n = 1

\(T(b+1) = \frac{(b+1)(b+2)}{2}=\frac{b^2+3b+2}{2}\)

\(T(b) = \frac{b(b+1)}{2}=\frac{b^2+b}{2}\)

\(T(b+1)-T(b)=b+1\)

\(b+1\) should be a triangular number.

The smallest triangular number after 2011 is 2016 (take 2011, double it, square root the result, round that, plug it in as n. It will at the very least get you close to the triangular number wanted and then you will have know that number is the nth triangular number, making finding the next easy).

\(b+1=2016\)

\(b=2015\)

Note that \((a - 1) + (b - 1) = a + b - 2\)

From Vieta's, we know that \(a + b = -{b \over a} = {7 \over 2} \)

So, \((a - 1) + (b - 1) = {7 \over 2} - 2 = \color{brown}\boxed{1.5}\)

\(\frac{(b+1)(b+2)}{2}-\frac{b(b+1)}{2}=\frac{x(x+1)}{2}\)

\(2(b+1)=x(x+1)\)

\(\frac{x(x+1)}{2}-1=b>2011 \)

So:

\(x(x+1)>4024\)

if x=63, then this satisfies the condition, and is the smallest positive integer x. (As x=62 won't be greater than 4024).

So:

\(2(b+1)=4032 \implies b=2015\) satisfies the condition.

Check:

\(T(2016)-T(2015)=2033136-2031120=2016=\frac{63(63+1)}{2}=T(63)\)

Answer: \(\mathbb{R} - (-1, 2)\) (with what I have, anyway...)

I assume this is what you mean: \((\frac{2-n}{n+1})+(\frac{2n-4}{2-n})\).

Of course, n could be literally anything other than -1 or 2 because there isn't an equality present (maybe you forgot it????? Or maybe I went blind????)

The answer is 2018.

The smallest integer that works is 4.

I'm with  BuilderBoi apart from one small slip.

3^96 * 3^384 = 3^480, not 3^470.

That gets  the result  a + b + c = -392.

Since probabilities must be less than 1, I've interpreted the question as follows:

Here's my take on this:

AH : HD   ≈  2.59 :  1.08

2x^2 - 7x  + 2   = 0

sum of roots  =  7/2  = 3.5

product of roots  = 2/2  = 1

(a - 1) (b -1)  =  ab - (a+ b) + 1 =   1 - (3.5) + 1  =   -1.5

l + f = 5/4l

l - 20 = 8/15 * 5/4l

l - 20 = 2/3l

3 (l - 20) = 2l

3l - 60 = 21

3l - 21 = 60

l = 60

l + f = 5/4l

f = 5/4l - l

f = (5l - 4l)/4

f = l/4, f = 60/4, f = 15

l = 60 and f = 15

The roots of the quadratic are \({15 \pm \sqrt{225-4t} \over 2}\)

Note that \(\sqrt {225 - 4t}\) must be an odd, positive integer. 

To solve, set \(\sqrt{225 - 4t}\) equal to 13, 11, 9, 7, 5, 3, and 1.

Now, solve for t in all cases and find the average.

Simplify: \(x^2 + y^2 + 5x - 3y - 24 = 0 \)

Add \(1.5^2 \) and \(2.5^2\) to both sides to complete the square: \(x^2 + y^2 + 5x - 3y - 24 + 1.5^2 + 2.5^2 = 1.5^2 + 2.5^2\)

Now, add 24 to both sides: \(x^2 + y^2 + 5x - 3y + 1.5^2 + 2.5^2 = 24 + 1.5^2 + 2.5^2\)

Complete the square on x: \((x+2.5)^2 + y^2 - 3y - 24 = 1.5^2 + 2.5^2\)

Complete the square on y: \((x+2.5)^2 + (y-1.5)^2 = 1.5^2 +2.5^2 + 24\)

Simplify the right-hand side: \((x+2.5)^2 + (y - 1.5)^2 = 32.5\)

The radius of this circle is \(\sqrt{32.5}\), so the area is \(\color{brown}\boxed{32.5 \pi}\)

ACB forms a right triangle in the circle

We have something like this : 

By the Theory of Intersecting Chords

AD * BD  =  CD * MD

5.9 * 3.04  = 3.88 * MD

(5.9)(3.04) / 3.88  = MD  ≈  4.62

So

CM =  CD + MD  =  3.88 + 4.62  ≈   8.5

Given that a > 0, if f(g(a)) = 8, where f(x) = x^2 + 4 and g(x) = x^2 - 4, what is the value of a?

g(a)  =  (a^2 - 4)

f(g(a))  =   (a^2 - 4)^2 + 4  =   8

(a^2 - 4)^2  =  8 - 4

(a^2 -4)^2  = 4              take the square root

a^2  - 4 =  ± 2

a^2 - 4  = 2               or              a^2  - 4  =  -2

a^2 = 6                                      a^2  =  2

a = sqrt (6)                                 a = sqrt (2)

(x^3  + 8) ( x + 2)   =  19

x^4 + 2x^3 + 8x - 3   =  0

x ≈ .36110

a + b =  7                square both sides

a^2  + 2ab +b^2  = 49

a^2 + b^2  = 49  - 2ab

And

a^3 + b^3  =  44

(a + b) ( a^2  - ab + b^2)  = 44

(a + b)  ( a^2 + b^2  - ab)  = 44

(7) ( 49 - 2ab -ab)  =  44

49 - 3ab  =  44/7

49 - 44/7  = 3ab

299/7  =  3ab

299 / 21  =  ab

Note that    1/a + 1/b =  (a + b) /  ab   =    7 / ( 299/21)  = 147 / 299

Set the equations equal

x^2 - 3x - 3 =  -x^2  - x  + 1              rearrange as

2x^2 - 2x - 4   =  0              divide through by 2

x^2 - x  - 2   =  0

(x - 2) ( x +1)   = 0

x -  2   = 0                 x + 1  = 0

x  = 2  = c                 x =  -1  =   a

c - a =   2 -  (- 1 )  =   3

x^2 + y^2   = 1        square both sides

x^4 + 2x^2y^2  + y^2  =  1

(x^4 + y^4)  +  2(xy)^2    =1

(17/19)  + 2(xy)^2   =  1

2(xy)^2  =   1 -  17/19

2(xy)^2   = 2/19             divide through by 2

(xy)^2  =    1/19            take the positive root

xy  = sqrt (1/19)  =   sqrt (19) / 19