POW will resume on Apr. 29.

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?

I hope you noticed the day this problem appeared was April fool’s day. However, we sincerely apologize to the students got confused about the problem description, and we found that many students already submitted the solution corresponding to the original problem.

Hence We revise the problem as the following:

Show that there do not exist perfect squares a, b, c such that \(a^2 + b^2 = c^2\), provided that a, b, c are nonzero integers.

You should actually provide the full valid proof i.e. the solution like ‘It is the special case of some famous theorem hence it is trivial’ will not be graded. Please resubmit your solution if you already submitted the solution for the previous version.

We accept the solution until April 11 Monday, 6PM.

Show that there exist perfect squares a, b, c such that \(a^2 + b^2 = c^2\).

====== REVISED (2022-04-04) ======

I hope you noticed the day this problem appeared was April fool’s day. Show instead that there do not exist perfect squares a, b, c such that \(a^2 + b^2 = c^2\), provided that a, b, c are nonzero integers.

For \(k,n\geq 1\), let \(v_1,\dots, v_n\) be unit vectors in \(\mathbb{R}^k\). Prove that we can always choose signs \(\varepsilon_1,\dots,\varepsilon_n\in \{-1, +1\}\) such that \(|\sum_{i=1}^{n} \varepsilon_i v_i |\leq \sqrt{n} \).

For any positive integer \(n \geq 2\), let \(B_n\) be the group given by the following presentation\[ B_n = < \sigma_1, \ldots, \sigma_{n-1} | \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}, \sigma_i \sigma_j = \sigma_j \sigma_i > \]where the first relation is for \( 1 \leq i \leq n-2 \) and the second relation is for \(|i-j| \geq 2\). Show that there exists a total order < on \(B_n\) such that for any three elements \(a, b, c\in B_n\), if \(a < b\) then \(ca < cb\). 

There are \(n\) people participating to a chess tournament and every two players play one game. There are no draws. Let \(a_i\) be the number of wins of the \(i\)-th player and \(b_i\) be the number of losses of the \(i\)-th player. Prove that
\[\sum_{i\in [n]} a_i^2 = \sum_{i\in [n]} b_i^2.\]

Let \(n, m\) be positive integers where \(m\) divides \(n\). When there exists a regular \(n\)-gon with area 1, what is the area of the largest regular \(m\)-gon inscribed in the \(n\)-gon in terms of \(n\) and \(m\)?