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.
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.
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.
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.
A real sequence \( x_1, x_2, x_3, \cdots \) satisfies the relation \( x_{n+2} = x_{n+1} + x_n \) for \( n = 1, 2, 3, \cdots \). If a number \( r \) satisfies \( x_i = x_j = r \) for some \( i \) and \( j \) \( (i \neq j) \), we say that \( r \) is a repeated number in this sequence. Prove that there can be more than \( 2013 \) repeated numbers in such a sequence, but it is impossible to have infinitely many repeated numbers.
The best solution was submitted by 진우영. Congratulations!
Similar solutions are submitted by 김범수(+3), 김홍규(+3), 김호진(+3), 남재현(+3), 박민재(+3), 박지민(+3), 박훈민(+3), 안현수(+3), 이시우(+3), 이주호(+3), 정성진(+3), 정우석(+3), 조정휘(+3), 진우영(+3). Thank you for your participation.
For real numbers \( a, b \), find the following limit.
\[
\lim_{n \to \infty} n \left( 1 – \frac{a}{n} – \frac{b \log (n+1)}{n} \right)^n.
\]
The best solution was submitted by 박민재. Congratulations!
Similar solutions are submitted by 김범수(+3), 박훈민(+3), 장경석(+3), 정성진(+3), 진우영(+3), 김홍규(+2), 박경호(+2). Thank you for your participation.
Let \( x, y \) be real numbers satisfying \( y \geq x^2 + 1 \). Prove that there exists a bounded random variable \( Z \) such that
\[
E[Z] = 0, E[Z^2] = 1, E[Z^3] = x, E[Z^4] = y.
\]
Here, \( E \) denotes the expectation.
The best solution was submitted by 정성진. Congratulations!
Other solutions are submitted by 박민재(+3), 이주호(+3), 장경석(+3), 진우영(+3). Thank you for your participation.
Let \(A, B\) are \(N \times N \) complex matrices satisfying \( rank(AB – BA) = 1 \). Prove that \( (AB – BA)^2 = 0 \).
The best solution was submitted by 김호진. Congratulations!
Similar solutions were submitted by 강동엽(+3), 김범수(+3), 김홍규(+3), 박민재(+3), 박지민(+3), 박훈민(+3), 안가람(+3), 어수강(+3), 엄문용(+3), 유찬진(+3), 이성회(+3), 이시우(+3), 이주호(+3), 장경석(+3), 전한솔(+3), 정동욱(+3), 정성진(+3), 정종헌(+3), 정우석(+3), 진우영(+3), Fardad Pouran(+3). Thank you for your participation.
Find all continuous functions \(f : \mathbb{R} \to \mathbb{R}\) satisfying
\[
f(x) = f(x^2 + \frac{x}{3} + \frac{1}{9} )
\]
for all \( x \in \mathbb{R} \).
The best solution was submitted by 강동엽. Congratulations!
Similar solutions were submitted by 김기현(+3), 김범수(+3), 김정섭(+3), 김호진(+3), 김홍규(+3), 박민재(+3), 박지민(+3), 박훈민(+3), 어수강(+3), 엄문용(+3), 윤성철(+3), 이명재(+3), 이성회(+3), 이시우(+3), 이주호(+3), 장경석(+3), 전한솔(+3), 정동욱(+3), 정성진(+3), 정종헌(+3), 조정휘(+3), 진우영(+3), 안가람(+2), 박경호(+2), 정우석(+2). Thank you for your participation.
Remark 1. As written in the rules, please submit the solution by 12PM on Wednesday. Any solution submitted after 12PM will not be graded.
Remark 2. Please write your name in the solution (not just in the email).
Let \( A = \{ (a_1, a_2, \cdots, a_n : a_i = \pm 1 \, (i = 1, 2, \cdots, n) \} \subset \mathbb{R}^n \). Prove that, for any \( X \subset A \) with \( |X| > 2^{n+1}/n \), there exist three distinct points in \( X \) that are the vertices of an equilateral triangle.
The best solution was submitted by 서기원, 09학번. Congratulations!
Similar solutions were submitted by 라준현(08학번, +3), 김호진(09학번, +3), 황성호(13학번, +3), 박정현(일반, +3), 정요한(서울시립대, +3). Thank you for your participation.