1992 APMO 第 3 题

Let $n$ be an integer such that $n>3$. Suppose that we choose three numbers from the set $\{1,2, \ldots, n\}$. Using each of these three numbers only once and using addition, multiplication, and parenthesis, let us form all possible combinations.

(a) Show that if we choose all three numbers greater than $n / 2$, then the values of these combinations are all distinct.

(b) Let $p$ be a prime number such that $p \leq \sqrt{n}$. Show that the number of ways of choosing three numbers so that the smallest one is $p$ and the values of the combinations are not all distinct is precisely the number of positive divisors of $p-1$.

令 $n$ 为整数，使得 $n>3$。假设我们从集合 $\{1,2, \ldots, n\}$ 中选择三个数字。这三个数字中的每一个仅使用一次，并使用加法、乘法和括号，让我们形成所有可能的组合。

(a) 证明如果我们选择所有三个大于 $n / 2$ 的数字，那么这些组合的值都是不同的。

(b) 令 $p$ 为质数，使得 $p \leq \sqrt{n}$。证明选择三个数字使得最小的一个是$p$并且组合的值不完全不同的方式的数量正是$p-1$的正因数的数量。