2024 IMO Shortlist N5

Let $S$ be a finite nonempty set of prime numbers. Let $1=b_1<b_2<\cdots$ be the sequence of all positive integers whose prime divisors all belong to $S$. Prove that, for all but finitely many positive integers $n$, there exist positive integers $a_1,a_2,\ldots,a_n$ such that

$$

\frac{a_1}{b_1}+\frac{a_2}{b_2}+\cdots+\frac{a_n}{b_n}

=

\left\lceil\frac1{b_1}+\frac1{b_2}+\cdots+\frac1{b_n}\right\rceil.

$$

设 $S$ 为有限非空素数集。令 $1=b_1<b_2<\cdots$ 为所有素因数均属于 $S$ 的正整数按从小到大排列得到的数列。证明除有限多个正整数 $n$ 外，都存在正整数 $a_1,a_2,\ldots,a_n$，使得

$$

\frac{a_1}{b_1}+\frac{a_2}{b_2}+\cdots+\frac{a_n}{b_n}

=

\left\lceil\frac1{b_1}+\frac1{b_2}+\cdots+\frac1{b_n}\right\rceil。

$$