Find all integers \( n \) such that the following holds:
There exists a set of \( 2n \) consecutive squares \( S = \{ (m+1)^2, (m+2)^2, \dots, (m+2n)^2 \} \) (\( m \) is a nonnegative integer) such that \( S = A \cup B \) for some \( A \) and \( B \) with \( |A| = |B| = n \) and the sum of elements in \( A \) is equal to the sum of elements in \( B \).
Nubjook and Kai are playing a game. First, they take an empty 2019×2019 matrix and take turns to write numbers in each entry. Once the matrix is completely filled, Nubjook wins if the determinant is nonzero and Kai wins otherwise. If Nubjook does the first move, who has a winning strategy?
Let \( n \in \mathbb{Z}^+ \) and \( x, y \in \mathbb{R}^+ \) such that \( x^n + y^n = 1 \). Prove that
\[
(1-x)(1-y) \left( \sum_{k=1}^n \frac{1+x^{2k}}{1+x^{4k}} \right) \left( \sum_{k=1}^n \frac{1+y^{2k}}{1+y^{4k}} \right) < \frac{7}{10}. \]
The best solution was submitted by 하석민 (수리과학과 2017학번). Congratulations!
Here is his solution of problem 2019-17.
Another solution was submitted by 채지석 (수리과학과 2016학번, +3).
Let \( n \in \mathbb{Z}^+ \) and \( x, y \in \mathbb{R}^+ \) such that \( x^n + y^n = 1 \). Prove that
\[
(1-x)(1-y) \left( \sum_{k=1}^n \frac{1+x^{2k}}{1+x^{4k}} \right) \left( \sum_{k=1}^n \frac{1+y^{2k}}{1+y^{4k}} \right) < \frac{7}{10}.
\]
POW will resume on Nov. 1.
Suppose a group \(G\) has a finite index subgroup that maps onto the free group of rank 2. Show that every countable group can be embedded in one of the quotient groups of \(G\).
The best solution was submitted by 하석민 (수리과학과 2017학번). Congratulations!
Here is his solution of problem 2019-16.
Suppose a group \(G\) has a finite index subgroup that maps onto the free group of rank 2. Show that every countable group can be embedded in one of the quotient groups of \(G\).
Let \( A, B \) be \( n \times n \) Hermitian matrices. Find all positive integer \( n \) such that the following statement holds:
“If \( AB – BA \) is singular, then \( A \) and \( B \) have a common eigenvector.”
The best solution was submitted by 채지석 (수리과학과 2016학번). Congratulations!
Here is his solution of problem 2019-14.
A similar solution was submitted by 하석민 (수리과학과 2017학번, +3). Late solutions are not graded.
Let \( A, B \) be \( n \times n \) Hermitian matrices. Find all positive integer \( n \) such that the following statement holds:
“If \( AB – BA \) is singular, then \( A \) and \( B \) have a common eigenvector.”
A group \(G\) is called residually finite if for any nontrivial element \(g\) of \(G\), there exists a finite group \(K\) and a surjective homomorphism \(\rho: G \to K\) such that \(\rho(g)\) is a nontrivial element of \(K\).
Suppose \(G\) is a finitely generated residually finite group. Show that any surjective homomorphism from \(G\) to itself is an isomorphism.
The best solution was submitted by 채지석 (수리과학과 2016학번). Congratulations!
Here is his solution of problem 2019-14.
Other solutions were submitted by 김동률 (수리과학과 2015학번, +3), 김태균 (수리과학과 2016학번, +3), 하석민 (수리과학과 2017학번, +3).