1989 USAMO 第 1 题

For each positive integer $n$, let

$$$$
\begin{aligned} S_n &= 1 + \frac 12 + \frac 13 + \cdots + \frac 1n \\ T_n &= S_1 + S_2 + S_3 + \cdots + S_n \\ U_n &= \frac{T_1}{2} + \frac{T_2}{3} + \frac{T_3}{4} + \cdots + \frac{T_n}{n+1}. \end{aligned}
$$$$

Find, with proof, integers $0 < a,\ b,\ c,\ d < 1000000$ such that $T_{1988} = a S_{1989} - b$ and $U_{1988} = c S_{1989} - d$.

对于每个正整数 $n$，让

$$

$$

\begin{对齐} S_n &= 1 + \frac 12 + \frac 13 + \cdots + \frac 1n \\ T_n &= S_1 + S_2 + S_3 + \cdots + S_n \\ U_n &= \frac{T_1}{2} + \frac{T_2}{3} + \frac{T_3}{4} + \cdots + \frac{T_n}{n+1}。 \结束{对齐}

$$

$$

通过证明，找到整数 $0 < a,\ b,\ c,\ d < 1000000$，使得 $T_{1988} = a S_{1989} - b$ 和 $U_{1988} = c S_{1989} - d$。