# Concluding Fall 2013

This semester, we have several ties including 3 perfect scorers. The top 5 participants of the semester are:

• T-1st: 박민재 (11학번): 35 points
• T-1st: 진우영 (12학번): 35 points
• T-1rd: 정성진 (13학번): 35 points
• T-4th: 김호진 (09학번): 25 points
• T-4th: 박훈민 (13학번): 25 points

Hearty congratulations to the prize winners!

We thank all of the participants for the nice solutions and your interest you showed for POW. We hope to see you next semester with better problems.

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# Solution: 2013-23 Polynomials with rational zeros

Find all polynomials $$P(x) = a_n x^n + \cdots + a_1 x + a_0$$ satisfying (i) $$a_n \neq 0$$, (ii) $$(a_0, a_1, \cdots, a_n)$$ is a permutation of $$(0, 1, \cdots, n)$$, and (iii) all zeros of $$P(x)$$ are rational.

The best solution was submitted by 전한솔. Congratulations!

Similar solutions are submitted by 김호진(+3), 박민재(+3), 엄태현(+3), 정성진(+3), 진우영(+3), 한대진(+3). Thank you for your participation.

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# 2013-23 Polynomials with rational zeros

Find all polynomials $$P(x) = a_n x^n + \cdots + a_1 x + a_0$$ satisfying (i) $$a_n \neq 0$$, (ii) $$(a_0, a_1, \cdots, a_n)$$ is a permutation of $$(0, 1, \cdots, n)$$, and (iii) all zeros of $$P(x)$$ are rational.

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# Solution: 2013-22 Field automorphisms

Find all field automorphisms of the field of real numbers $$\mathbb{R}$$. (A field automorphism of a field $$F$$ is a bijective map $$\sigma : F \to F$$ that preserves all of $$F$$’s algebraic properties.)

The best solution was submitted by 박지민. Congratulations!

Similar solutions are submitted by 고진용(+3), 김호진(+3), 박경호(+3), 박민재(+3), 박훈민(+3), 어수강(+3), 전한솔(+3), 정성진(+3), 진우영(+3), 한대진(+3). Thank you for your participation.

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Let $$f(z) = z + e^{-z}$$. Prove that, for any real number $$\lambda > 1$$, there exists a unique $$w \in H = \{ z \in \mathbb{C} : \text{Re } z > 0 \}$$ such that $$f(w) = \lambda$$.