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.
The best solution was submitted by 강한필 (전산학부 2016학번, +4). Congratulations!
Here is his solution of problem 2021-02.
Other solutions was submitted by 하석민 (수리과학과 2017학번, +3), 박은아 (수리과학과 2015학번, +2).
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.
The best solution was submitted by 강한필 (전산학부 2016학번, +4). Congratulations!
Here is his solution of problem 2021-01.
Other solutions was submitted by 김기수 (수리과학과 2018학번), 박은아 (수리과학과 2015학번, +3), 전해구 (기계공학과 졸업생).
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.