서울대학교 심층구술면접의 경우 답안이 따로 나오지 않아 제가 시간날때마다 재미있는 것들 위주로 올려보고자 합니다:) 문제는 아래와 같습니다:

Translation of the problem is as follows:

Problem.

For a rational $x$ greater than 1, if it decomposes as

$$ x=b_1-\frac{1}{b_2-\frac{b_3}{1-\cdots1-\frac{1}{b_s}}} $$

Let’s write $x=<b_1,b_2,\cdots,b_k>$ and call the staircase of $x$. For example, $25/9=<3,5,2>$.

3.1. For naturals $p>q$ show that the staircase $<b_1,b_2,\cdots,b_k>$ always exists.

3.2. For a natural $p>1$, find the staircase of $p^2/(p-1)$.

3.3. For the staircase you found in (3.2), suppose there are rationals $q_0,q_1,\cdots,q_{s+1}$ satisfying the following conditions:

(i) $q_0=2, q_{s+1}=1$

(ii) $q_{i-1}+q_{i+1}=b_iq_i, (1\leq i \leq s)$

Find $q_1$.

Solutions.

Below is my solution.