ElectricPavlov

Here is one possibility 

  your mom gives birth to your sister when your mom is 30    .... you are 7 at the time 

      10 years later, your sister is 10, your mom is 40   and you are 17 

Integers are   x     x+1     x+ 2    and  x+3 

summed they equal 18 

x  +    x+1   +   x+2   + x+3  = 18 

4x + 6 = 18 

4x = 12 

x = 3 

Nice work, The3Mathketeers ! 

18 P 3 = 18! / 15! = 4896 ways

18 choices for gold    17 choices for silve 16 choices for bronaze =   18 x 17 x 16 = 4896 ways 

Note that the number of Americans is irrelavent.

https://web2.0calc.com/questions/help-plz_89382#r1  

Arrange the given parabola equation to vertex (h,k)    form:  y =  a ( x- h) ^2 + k   

          by completing the square for x :

                 1/4 ( x^2 + 4x) +2

                 1/4 ( x^2 + 4x + 4)   -1 + 2

                 1/4 ( x+2)^2 +1      Shows vertex = -2,1   

 the equation of a parabola in vertex form y=a(x−h)^2+k, ======> the focus is (h,k+1/4a)

               focus is then  -2 , 1 + 1/(4 * 1/4) =   (  -2,2 )  

    and directrix is equal distance from the vertex as the focus and is    y = 0 

Each number is  2 times the previous number   minus 1  

  2 x 17 = 34     minus 1 = 33 

(31+x) = 2 ( 8 + x)        Given   where x is a number of years ....you age is 8+x 

31+ x = 16 + 2x 

x = 15           your age = 8 + 15 = 23 y/o