Answer: A

Small Hint:

Two dates fall on the same weekday exactly when the number of days between them is a multiple of $7$

Big Hint:

Use the two given Tuesdays to decide which of $N - 1, N, N + 1$ is a leap year

Solution:

From day $300$ of year $N$ to day $200$ of year $N + 1,$ the number of days is $365 - 300 + 200 = 265$ if $N$ is not a leap year. But $265 = 7 \cdot 37 + 6,$ which would land on a Monday, not a Tuesday.

So year $N$ is a leap year, and the gap is $266 = 7 \cdot 38$ days, giving a Tuesday as stated. It follows that year $N - 1$ is not a leap year.

The $100$th day of year $N - 1$ precedes the Tuesday on day $300$ of year $N$ by $365 - 100 + 300 = 565$ days. Since $565 = 7 \cdot 80 + 5,$ that day is $5$ weekdays before Tuesday, namely Thursday.

Thus, the correct answer is A.