Tag Archives: 양해훈

Solution: 2009-16 Commutative ring

Let k>1 be a fixed integer. Let π be a fixed nonidentity permutation of {1,2,…,k}. Let I be an ideal of a ring R such that for any nonzero element a of R, aI≠0 and Ia≠0 hold.

Prove that if \(a_1 a_2\ldots a_k=a_{\pi(1)} a_{\pi(2)} \ldots a_{\pi(k)}\)  for any elements \(a_1, a_2,\ldots,a_k \in I\), then R is commutative.

The best solution was submitted by Yang, Hae Hun (양해훈), 2008학번. Congratulations!

Here is his Solution of Problem 2009-16.

An alternative solution was submitted by 정성구(수리과학과 2007학번, +3).

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Solution:2008-11 Sum of square roots

Let a, b, c, d be positive rational numbers. Prove that if \(\sqrt a+\sqrt b+\sqrt c+\sqrt d\) is rational, then each of \(\sqrt a\), \(\sqrt b\), \(\sqrt c\), and \(\sqrt d\) is rational.

The best solution was submitted by Yang, Hae Hun (양해훈), 2008학번. Congratulations!

Here is his Solution of Problem 2008-11.

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Solution: 2008-9 Integer-valued function

Let \(\mathbb{R}\) be the set of real numbers and let \(\mathbb{N}\) be the set of positive integers. Does there exist a function \(f:\mathbb{R}^3\to \mathbb{N}\) such that f(x,y,z)=f(y,z,w) implies x=y=z=w?

The best solution was submitted by Yang, Hae Hun  (양해훈), 2008학번. Congratulations!

Here is his Solution of Problem 2008-9.

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Solution: 2008-6 Many primes

Let f(x) be a polynomial with integer coefficients. Prove that if f(x) is not constant, then there are infinitely many primes p such that \(f(x)\equiv 0\pmod p\) has a solution x.

The best solution was submitted by Yang, Hae Hun  (양해훈), 2008학번. Congratulations!

Here is his Solution of Problem 2008-6.

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