# 2021-04 Product of matrices

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 $$A, B$$,

$[\lambda(A^m B^m)]^{1/m} \leq [\lambda(A^{m+1} B^{m+1})]^{1/(m+1)}.$

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# Solution: 2021-03 A placement of rooks on a chessboard

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).

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# 2021-03 A placement of rooks on a chessboard

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.

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# Solution: 2021-02 Inscribed triangles

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).

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# Solution: 2021-01 Single-digit number

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).

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# 2021-02 Inscribed triangles

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.

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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).