When is my birthday? 
Ariane, Gian and I have unique birthdays.
  We do not have the same birthday yet we were born on the same day. It is four less than the sum of our birth months. The sum of our ages as of now is 18 less than thrice the sum of our months. None of us belong on the same year but it is one after another as well as our months.
  When I tried adding my age and twice the day we were born, I got 66.
  The last to celebrate his/her birthday is one year older than me and the first to celebrate is the oldest.

I want to check my answer, am I correct. My working got me that my birthday is 23 october 1995. Am I correct?
 A: Let $d$ be the day of birth. The birth months are consecutive, so if $m$ denotes the middle month, then

we were born on the same day. It is four less than the sum of our birth months

implies
$d=(m-1)+m+(m+1)-4 = 3m-4$. Let $\Sigma a$ denote the sum of our ages now. Then

The sum of our ages as of now is 18 less than thrice the sum of our months

implies $\Sigma a=9m-18$.  If $\bar a$ denotes the average of our ages now, then $\bar a=3m-6$. Since our birth years are consecutive, my age can't be far from $\bar a$. It must be $\bar a +\epsilon$ where $\epsilon$ is some small number, no larger in absolute value than, say $3$. Since

When I tried adding my age and twice the day we were born, I got 66

it follows that $\bar a + \epsilon + 2d = 66$, or $$3m-6+\epsilon+2(3m-4)=66\quad\Longleftrightarrow\quad9m+\epsilon=80$$
The only value of $m$ that satisfies is $m=9$. So the birth months are 8,9,10, and $d=3m-4=23$. Plugging $m=9$ back into $\Sigma a = 9m-18$, we get $\Sigma a = 63$. Since none of our birthdays have happened as of today (June 12, 2015) and since our birth years are consecutive, our ages today must be 20, 21, 22.
Returning to the statement

When I tried adding my age and twice the day we were born, I got 66

my age today must be 20, so I am the youngest, born in 1994 (since I'll be 21 later this year). Finally,

The last to celebrate his/her birthday is one year older than me and the first to celebrate is the oldest

so I am neither the last nor the first to celebrate, so my birth month is the middle one, i.e., September. My birthday is Sept 23, 1994.
