# 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!

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# 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!

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# 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.

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# 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.

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# 2013-16 Limit of a sequence

For real numbers $$a, b$$, find the following limit.
$\lim_{n \to \infty} n \left( 1 – \frac{a}{n} – \frac{b \log (n+1)}{n} \right)^n.$

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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.