We have an expression \(x_0 \div x_1 \div x_2 \div \dots \div x_n\). A way of putting \(n-1\) left parentheses and \(n-1\) right parenthese is called ‘parenthesization’ if it is valid and completely clarify the order of operations. For example, when \(n=3\), we have the following five parenthesizations.
\[ ((x_0\div x_1)\div x_2)\div x_3, \enspace (x_0\div (x_1\div x_2))\div x_3, \enspace (x_0\div x_1)\div (x_2\div x_3),\]
\[x_0\div ((x_1\div x_2)\div x_3), \enspace x_0\div (x_1\div (x_2\div x_3)).
\]

(a) For an integer \(n\), how many parenthesizations are there?
(b) For each parenthesization, we can compute the expression to obtain a fraction with some variables in the numerator and some variable in the denominator. For an integer \(n\), determine which fraction occur most often. How many times does it occur?

The best solution was submitted by 나영준 (연세대학교 의학과 18학번, +4). Congratulations!

Here is the best solution of problem 2022-06.

Other (incomplete) solutions were submitted by 조유리 (문현여고 3학년, +2), 이명규 (KAIST 전산학부 20학번, +2), 박기찬 (KAIST 새내기과정학부 22학번, +2), Antonio Recuero Buleje (+2).