For given \(k\in \mathbb{N}\), determine the minimum natural number \(n\) satisfying the following: no matter how one colors each number in \(\{1,2,\dots, n\}\) red or blue, there always exists (not necessarily distinct) numbers \(x_0, x_1,\dots, x_k \in [n]\) with the same color satisfying \(x_1+\dots + x_k = x_0\).
Prove or disprove that if C is any nonempty connected, closed, self-antipodal (ie., invariant under the antipodal map) set on \(S^2\), then it equals the zero locus of an odd, smooth function \(f:S^2 -> \mathbb{R}\).
Let \( A_N \) be an \( N \times N \) matrix whose entries are i.i.d. Bernoulli random variables with probability \( 1/2 \), i.e.,
\[\mathbb{P}( (A_N)_{ij} =0) = \mathbb{P}( (A_N)_{ij} =1) = \frac{1}{2}.\]
Let \( p_N \) be the probability that \( \det A_N \) is odd. Find \( \lim_{N \to \infty} p_N \).
Let \(\mathcal{A}_n\) be the collection of all sequences \( \mathbf{a}= (a_1,\dots, a_n) \) with \(a_i \in [i]\) for all \(i\in [n]=\{1,2,\dots, n\}\). A nondecreasing \(k\)-subsequence of \(\mathbf{a}\) is a subsequence \( (a_{i_1}, a_{i_2},\dots, a_{i_k}) \) such that \(i_1< i_2< \dots < i_k\) and \(a_{i_1}\leq a_{i_2}\leq \dots \leq a_{i_k}\). For given \(k\), determine the smallest \(n\) such that any sequence \(\mathbf{a}\in \mathcal{A}_n\) has a nondecreasing \(k\)-subsequence.
Prove or disprove that if all elements of an infinite group G has order less than n for some positive integer n, then G is finitely generated.
For an \( n \times n \) matrix \( M \) with real eigenvalues, let \( \lambda(M) \) be the largest eigenvalue of \( M\). Prove that for any positive integer \( r \) and positive semidefinite matrices \( A, B \),
\[[\lambda(A^m B^m)]^{1/m} \leq [\lambda(A^{m+1} B^{m+1})]^{1/(m+1)}.\]
Consider an \(n\) by \(n\) chessboard with white/black squares alternating on every row and every column. In how many ways can one choose \(k\) white squares and \(n-k\) black squares from this chessboard with no two squares in a row or column.
The best solution was submitted by 강한필 (전산학부 2016학번, +4). Congratulations!
Here is his solution of problem 2021-03.
Other solutions was submitted by 하석민 (수리과학과 2017학번, +3), 고성훈 (수리과학과 2015학번, +3), 전해구 (기계공학과 졸업생, +3).
Consider an \(n\) by \(n\) chessboard with white/black squares alternating on every row and every column. In how many ways can one choose \(k\) white squares and \(n-k\) black squares from this chessboard with no two squares in a row or column.
Show that for any triangle T and any Jordan curve C in the Euclidean plane, there exists a triangle inscribed in C which is similar to T.
Prove that for any given positive integer \( n \), there exists a sequence of the following operations that transforms \( n \) to a single-digit number (in decimal representation).
1) multiply a given positive integer by any positive integer.
2) remove all zeros in the decimal representation of a given positive integer.