There are $20$ balls in a bag. $a$ of them are red, $b$ of them are white, and $c$ of them are black. It is known that $ \bullet$ if we double the white balls, the probability of drawing one red ball is $\dfrac 1{25}$ less than the probability of drawing one red ball at the beginning, and $ \bullet$ if we remove all red balls, the probability of drawing one white ball is $\dfrac 1{16}$ more than the probability of drawing one white ball at the beginning. Find $a,b,c$.

Write out the positive integers consisting of only $1$s, $6$s, and $9$s in ascending order as in: $1,6,9,11,16,\dots$. a. Find the order of $1996$ in the sequence. b. Find the $1996$th term in the sequence.

Let $P$ be a point inside of equilateral $\triangle ABC$ such that $m(\widehat{APB})=150^\circ$, $|AP|=2\sqrt 3$, and $|BP|=2$. Find $|PC|$.