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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Solution: 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 $$n$$.

The best solution was submitted by 박기찬 (KAIST 새내기과정학부 22학번, +4). Congratulations!

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