2020/21 AHSMC Part 1 Problem 13

Let $n$ be the number of groups, and let us order the groups increasingly by the number of marbles.

The first group has at least $1$ marble, the second group has at least $2$ marbles, $\ldots$, the last group has at least $n$ marbles. In total she has at least $1+2+3+\cdots+n = \dfrac{n(n+1)}{2}$ marbles. Therefore, $$\frac{n(n+1)}{2} \le 200.$$ By inspection we see that $n = 20$ is too large, but $n = 19$ is the maximal solution.

$n = 19$ is an acceptable solution, as she could have groups of $1, 2, 3, \ldots, 18$ and $29$ marbles. The answer is (A).