# 2022-12 A partition of the power set of a set

Consider the power set $$P([n])$$ consisting of $$2^n$$ subsets of $$[n]=\{1,\dots,n\}$$.
Find the smallest $$k$$ such that the following holds: there exists a partition $$Q_1,\dots, Q_k$$ of $$P([n])$$ so that there do not exist two distinct sets $$A,B\in P([n])$$ and $$i\in [k]$$ with $$A,B,A\cup B, A\cap B \in Q_i$$.

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# 2022-11 groups with torsions

Does there exists a finitely generated group which contains torsion elements of order p for all prime numbers p?

Solutions for POW 2022-11 are due July 4th (Saturday), 12PM, and it will remain open if nobody solved it.

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# 2022-10 Polynomial with root 1

Prove or disprove the following:

For any positive integer $$n$$, there exists a polynomial $$P_n$$ of degree $$n^2$$ such that

(1) all coefficients of $$P_n$$ are integers with absolute value at most $$n^2$$, and

(2) $$1$$ is a root of $$P_n =0$$ with multiplicity at least $$n$$.

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# 2022-09 A chaotic election

Let $$A_1,\dots, A_k$$ be presidential candidates in a country with $$n \geq 1$$ voters with $$k\geq 2$$. Candidates themselves are not voters. Each voter has her/his own preference on those $$k$$ candidates.

Find maximum $$m$$ such that the following scenario is possible where $$A_{k+1}$$ indicates the candidate $$A_1$$: for each $$i\in [k]$$, there are at least $$m$$ voters who prefers $$A_i$$ to $$A_{i+1}$$.

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# 2022-08 two sequences

For positive integers $$n \geq 2$$, let $$a_n = \lceil n/\pi \rceil$$ and let $$b_n = \lceil \csc (\pi/n) \rceil$$. Is $$a_n = b_n$$ for all $$n \neq 3$$?

Solutions are due May 13th (Friday), 6PM, and it will remain open if nobody solved it.

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# 2022-07 Coulomb potential

Prove the following identity for $$x, y \in \mathbb{R}^3$$:
$\frac{1}{|x-y|} = \frac{1}{\pi^3} \int_{\mathbb{R}^3} \frac{1}{|x-z|^2} \frac{1}{|y-z|^2} dz.$

Solutions are due May 6th (Friday), 6PM, and it will remain open if nobody solved it.

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# Notice: Mid-term break

POW will resume on Apr. 29.

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# 2022-06 A way of putting parentheses

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?

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# Notice on POW 2022-05 (Problem Revision)

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.

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