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.)

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2013-21 Unique inverse

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 $$f(w) = \lambda$$.

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Solution: 2013-20 Eigenvalues of Hermitian matrices

Let $$A, B, C = A+B$$ be $$N \times N$$ Hermitian matrices. Let $$\alpha_1 \geq \cdots \geq \alpha_N$$, $$\beta_1 \geq \cdots \geq \beta_N$$, $$\gamma_1 \geq \cdots \geq \gamma_N$$ be the eigenvalues of $$A, B, C$$, respectively. For any $$1 \leq i, j \leq N$$ with $$i+j -1 \leq N$$, prove that
$\gamma_{i+j-1} \leq \alpha_i + \beta_j$

The best solution was submitted by 진우영. Congratulations!

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

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2013-20 Eigenvalues of Hermitian matrices

Let $$A, B, C = A+B$$ be $$N \times N$$ Hermitian matrices. Let $$\alpha_1 \geq \cdots \geq \alpha_N$$, $$\beta_1 \geq \cdots \geq \beta_N$$, $$\gamma_1 \geq \cdots \geq \gamma_N$$ be the eigenvalues of $$A, B, C$$, respectively. For any $$1 \leq i, j \leq N$$ with $$i+j -1 \leq N$$, prove that
$\gamma_{i+j-1} \leq \alpha_i + \beta_j$

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Solution: 2013-19 Integral inequality

Suppose that a function $$f:[0, 1] \to (0, \infty)$$ satisfies that
$\int_0^1 f(x) dx = 1.$
Prove the following inequality.
$\left( \int_0^1 |f(x)-1| dx \right)^2 \leq 2 \int_0^1 f(x) \log f(x) dx.$

The best solution was submitted by 정성진. Congratulations!

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

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2013-19 Integral inequality

Suppose that a function $$f:[0, 1] \to (0, \infty)$$ satisfies that
$\int_0^1 f(x) dx = 1.$
Prove the following inequality.
$\left( \int_0^1 |f(x)-1| dx \right)^2 \leq 2 \int_0^1 f(x) \log f(x) dx.$

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Solution: 2013-18 Idempotent elements

Let $$R$$ be a ring of characteristic zero. Assume further that $$na \neq 0$$ for a positive integer $$n$$ and $$a \in R$$ unless $$a = 0$$. Suppose that $$e, f, g \in R$$ are idempotent (with respect to the multiplication) and satisfy $$e + f + g = 0$$. Show that $$e = f = g = 0$$. (An element $$a$$ is idempotent if $$a^2 = a$$. )

The best solution was submitted by 박훈민. Congratulations!

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

Remark: Special thanks to 김동현, who first reported that the condition `characteristic zero’ is insufficient for the problem.

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Notice on 2013-18

Please notice that one more assumption is added in Problem 2013-18. (Due to a mistake) I apologize for the confusion and inconvenience.

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Let $$R$$ be a ring of characteristic zero. Assume further that $$na \neq 0$$ for a positive integer $$n$$ and $$a \in R$$ unless $$a = 0$$. Suppose that $$e, f, g \in R$$ are idempotent (with respect to the multiplication) and satisfy $$e + f + g = 0$$. Show that $$e = f = g = 0$$. (An element $$a$$ is idempotent if $$a^2 = a$$. )