Purple College keeps a careful count of its students as they progress each year from the freshman class to the sophomore class to the junior class and, finally, to the senior class.  Each year at the college one third of the freshman class drops out of school, $40$ students in the sophomore class drop out of school, and one tenth of the junior class drops out of school.  Given that the college only admits new freshman students, and that it wants to begin each school year with $3400$ students enrolled, how many students does it need to admit into the freshman class each year?
1. **Determine the number of students needed in the junior class at the beginning of the year:**
   - Given that one tenth of the junior class drops out, we need to start with enough students so that after the dropouts, there are 3400 students remaining.
   - Let \( J \) be the number of students in the junior class at the beginning of the year.
   - We have the equation:
     \[
     J - \frac{1}{10}J = 3400
     \]
   - Simplifying, we get:
     \[
     \frac{9}{10}J = 3400
     \]
   - Solving for \( J \):
     \[
     J = \frac{3400 \times 10}{9} = 3777.\overline{7}
     \]
   - Since the number of students must be an integer, we round up to the nearest whole number:
     \[
     J = \left\lceil 3777.7 \right\rceil = 3778
     \]

2. **Determine the number of students needed in the sophomore class at the beginning of the year:**
   - Given that 40 students drop out of the sophomore class, we need to start with enough students so that after the dropouts, there are 3778 students remaining.
   - Let \( S \) be the number of students in the sophomore class at the beginning of the year.
   - We have the equation:
     \[
     S - 40 = 3778
     \]
   - Solving for \( S \):
     \[
     S = 3778 + 40 = 3818
     \]

3. **Determine the number of students needed in the freshman class at the beginning of the year:**
   - Given that one third of the freshman class drops out, we need to start with enough students so that after the dropouts, there are 3818 students remaining.
   - Let \( F \) be the number of students in the freshman class at the beginning of the year.
   - We have the equation:
     \[
     F - \frac{1}{3}F = 3818
     \]
   - Simplifying, we get:
     \[
     \frac{2}{3}F = 3818
     \]
   - Solving for \( F \):
     \[
     F = \frac{3818 \times 3}{2} = 5727
     \]
   - Since the number of students must be an integer, we round up to the nearest whole number:
     \[
     F = \left\lceil 5727 \right\rceil = 5727
     \]

The final answer is \(\boxed{5727}\)