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!

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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!

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.

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.

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.

\]

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.