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\(1/(1 + 1/(sqrt(3) + 2))\\ =\frac{1}{1 + \frac{1}{(\sqrt3 + 2)}}\\ =1\div( 1 + \frac{1}{(\sqrt3 + 2)})\\ =1\div( \frac{\sqrt3+2+1}{(\sqrt3 + 2)})\\ =1\div \frac{3+\sqrt3}{2+\sqrt3 }\\\)

Invert and multiply

then

multiply top and bottom by the conjugate of the bottom.

Can you take it from there?

probably...\(x=-99+\sqrt{8021},\:c=0\)

Are you Tuckerrrr in disguise?

The answer is 21 because:

5x = 3

x = 3/5

5( 2(3/5) + 3)

5 ( 6/5 + 3)

6 + 15

21

8y^{11} + 6y^{9} - 2y^{6}

I believe there are seven solutions to this. These are listed below:

Each three numbers inside parentheses correspond to the values of a, b, and c in that order in a specific solution:

(0, 0, 2) (0, 3, 0) (1, 1, 1) (2, 2, 0) (3, 0, 1) (4, 1, 0) (6, 0, 0)

8km / 3cm   *  11 cm   =   29 1/3  cm     ( see how the units   cm   cancel out and you are left with 'km' ? 

From a Periodic Table

   carbon weighs ~ 12 gm/mole

   oxygen weighs ~ 16 gm/mole

       so CO₂   weighs  44 gm/mole

            214 kg = 214 000 gm

214 000 gm  /  44gm / mole = 4863.6 moles

4863.6 moles   *    6.02 x 10²³  particles/mole    = 2.928 x 10²⁷  particles of CO₂

...I give up

I solved this problem before.  All possible integers are 8, 15, and 24.

1. 14

2. 100a + 10b + c = 283

3. The length of line AB is 2*sqrt(6).

4. a + b + c = 17

6. The vertex is (4,14)

7. The equation of symmetry is x = 4.

8. a + b + c = 14.

That answer is wrong.

The above should read:

[2019 - 3] / 3  +  3=675 such fractions.

1 - This means: every 3 cm on the map ==8 km on the ground. So, 1 cm on the map==8 / 3==2+2/3 km on the ground. Therefore, 11 cm on the map ==11 cm x 2+2/3 km/cm ==29 + 1/3 km - actual distance between the 2 cities.

OR:

2 -  [11 cm / 3 cm] x 8 km /3 cm ==[3 + 2/3]  x  8 ==29+1/3 km - actual distance btween the two cities.

2^10 ==1024 subsets (numbers)

∅ | {1} | {2} | {3} | {4} | {5} | {11} | {12} | {13} | {14} | {15} | {1, 2} | {1, 3} | {1, 4} | {1, 5} | {1, 11} | {1, 12} | {1, 13} | {1, 14} | {1, 15} | {2, 3} | {2, 4} | {2, 5} | {2, 11} | {2, 12} | {2, 13} | {2, 14} | {2, 15} | {3, 4} | {3, 5} | {3, 11} | {3, 12} | {3, 13} | {3, 14} | {3, 15} | {4, 5} | {4, 11} | {4, 12} | {4, 13} | {4, 14} | {4, 15} | {5, 11} | {5, 12} | {5, 13} | {5, 14} | {5, 15} | {11, 12} | {11, 13} | {11, 14} | {11, 15} | {12, 13} | {12, 14} | {12, 15} | {13, 14} | {13, 15} | {14, 15} | {1, 2, 3} | {1, 2, 4} | {1, 2, 5} | {1, 2, 11} | {1, 2, 12} | {1, 2, 13} | {1, 2, 14} | {1, 2, 15} | {1, 3, 4} | {1, 3, 5} | {1, 3, 11} | {1, 3, 12} | {1, 3, 13} | {1, 3, 14} | {1, 3, 15} | {1, 4, 5} | {1, 4, 11} | {1, 4, 12} | {1, 4, 13} | {1, 4, 14} | {1, 4, 15} | {1, 5, 11} | {1, 5, 12} | {1, 5, 13} | ... (total: 1024)

1024 - 1 empty subset - 10 single-digit subsets ==1013  subsets, or different products.

If f(x)=ax when x=-4, then...

-60/13 = a * -4

15/13 = a 

If f(x)=bx when x>-4, then...

3120=b * 4

780 = b 

So, a+b = 15/13 + 780 = 10155/13.

Numerators that are multiples of three are able to express the fraction in lowest form, but aren't actually so expressed here.  For example, 3/2019 isn't in lowest form.

And what about numerators that are multiples of 673?