Please notice that one more assumption is added in Problem 2013-18. (Due to a mistake) I apologize for the confusion and inconvenience.
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 \( a^2 = a \). )
Midterm break
The problem of the week will take a break during the midterm period and return on Nov. 1, Friday. Good luck on your midterm exams!
Solution: 2013-17 Repeated numbers
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.
Solution: 2013-16 Limit of a sequence
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.
2013-17 Repeated numbers
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.
2013-16 Limit of a sequence
For real numbers \( a, b \), find the following limit.
\[
\lim_{n \to \infty} n \left( 1 – \frac{a}{n} – \frac{b \log (n+1)}{n} \right)^n.
\]
Solution: 2013-15 Bounded random variable
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.
2013-15 Bounded random variable
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.
Solution: 2013-14 Nilpotent matrix
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.