Largest possible value of first term

Let $S = a_1 + a_2 + \dots + a_{52}$. Given that the average of all 52 integers is 1 less than the average of $a_2, \dots, a_{52}$: $\frac{S}{52} = \frac{S - a_1}{51} - 1$ $51S = 52(S - a_1) - 52 \times 51$ $51S = 52S - 52a_1 - 2652$ $S = 52a_1 + 2652$

Now, $S = a_1 + (a_2 + a_3 + \dots + a_{52})$. To maximize $a_1$, we must minimize $a_2 + a_3 + \dots + a_{51}$ given $a_{52} = 100$ and $a_1 < a_2 < \dots < a_{52}$. For minimum sum, $a_2, a_3, \dots, a_{51}$ should be in consecutive increasing order ending just below $a_{52} = 100$. So $a_{51} = 99, a_{50} = 98, \dots, a_2 = 99 - 49 = 50$. The sum $a_2 + \dots + a_{52} = \frac{51}{2}(50 + 100) = 51 \times 75 = 3825$.

So $S = a_1 + 3825$. Using $S = 52a_1 + 2652$: $a_1 + 3825 = 52a_1 + 2652$ $51a_1 = 1173 \implies a_1 = 23$