# Solution: 2019-17 0.7?

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).

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# 2019-17 0.7?

Let $$n \in \mathbb{Z}^+$$ and $$x, y \in \mathbb{R}^+$$ such that $$x^n + y^n = 1$$. Prove that
$(1-x)(1-y) \left( \sum_{k=1}^n \frac{1+x^{2k}}{1+x^{4k}} \right) \left( \sum_{k=1}^n \frac{1+y^{2k}}{1+y^{4k}} \right) < \frac{7}{10}.$

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# Notice: Mid-term break

POW will resume on Nov. 1.

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# Solution: 2019-16 Groups with abundant quotients

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.

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# 2019-16 Groups with abundant quotients

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 $$G$$.

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# Solution: 2019-15 Singular matrix

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.

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# 2019-15 Singular matrix

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.”

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# Solution: 2019-14 Residual finite groups

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).

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# 2019-14 Residual finite groups

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.

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# Solution: 2019-12 Groups generated by two homeomorphisms of the real line

Let $$I, J$$ be connected open intervals such that $$I \cap J$$ is a nonempty proper sub-interval of both $$I$$ and$$J$$. For instance, $$I = (0, 2)$$ and $$J = (1, 3)$$ form an example.

Let $$f$$ ($$g$$, resp.) be an orientation-preserving homeomorphism of the real line $$\mathbb{R}$$ such that the set of points of $$\mathbb{R}$$ which are not fixed by $$f$$ ($$g$$, resp.) is precisely $$I$$ ($$J$$, resp.).

Show that for large enough integer $$n$$, the group generated by $$f^n, g^n$$ is isomorphic to the group with the following presentation

$<a, b | [ab^{-1}, a^{-1}ba] = [ab^{-1}, a^{-2}ba^2] = id>.$

The best solution was submitted by 김동률 (수리과학과 2015학번). Congratulations!

Here is his solution of problem 2019-12.

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