# 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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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.