Self-referential truth count

Let the statements be numbered $1,2,\dots,100$. Statement $k$ says: "at most $k-1$ of these $100$ statements are true."

Let $T$ be the total number of true statements. For statement $k$ to be true, we must have $T \le k-1$; for it to be false, $T > k-1$. Hence statement $k$ is true exactly when $k \ge T+1$.

Thus the true statements are those with indices $T+1, T+2, \dots, 100$. The number of such statements is $100 - T$. But this number must equal $T$ itself, because $T$ is the total number of true statements. Therefore:

Check: If $T=50$, then statements $51$ through $100$ are true (50 statements). Statement $51$ says "at most $50$ are true" — true because exactly $50$ are true. Statement $50$ says "at most $49$ are true" — false because $50 > 49$. All statements $1$ through $49$ are false because each claims "at most $k-1$ are true" with $k-1 \le 48$, but $50 > k-1$. All statements $52$ through $100$ are true because each claims "at most $k-1$ are true" with $k-1 \ge 51$, and $50 \le k-1$. Hence exactly $50$ statements are true, consistent with $T=50$.

Thus the answer is $50$.