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.
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.
Solution: 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 \( f(w) = \lambda \).
The best solution was submitted by 박민재. Congratulations!
Similar solutions are submitted by 김동률(+3), 김범수(+3), 김호진(+3), 박지민(+3), 박훈민(+3), 양지훈(+3), 이시우(+3), 전한솔(+3), 정성진(+3), 조정휘(+3), 진우영(+3), Koswara(+3), Harmanto(+3). Thank you for your participation.
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.)
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 \( f(w) = \lambda \).
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.
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
\[ \gamma_{i+j-1} \leq \alpha_i + \beta_j \]
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.
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.
\[ \left( \int_0^1 |f(x)-1| dx \right)^2 \leq 2 \int_0^1 f(x) \log f(x) dx. \]
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.