# 2017-18 Limit

Suppose that $$f$$ is differentiable and $\lim_{x\to\infty} (f(x)+f'(x))=2.$  What is $$\lim_{x\to\infty} f(x)$$?

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# 2017-17 An infimum

For an integer $$n \geq 3$$, evaluate
$\inf \left\{ \sum_{i=1}^n \frac{x_i^2}{(1-x_i)^2} \right\},$
where the infimum is taken over all $$n$$-tuple of real numbers $$x_1, x_2, \dots, x_n \neq 1$$ satisfying that $$x_1 x_2 \dots x_n = 1$$.

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# Solution: 2017-15 Infinite product

For $$x \in (1, 2)$$, prove that there exists a unique sequence of positive integers $$\{ x_i \}$$ such that $$x_{i+1} \geq x_i^2$$ and
$x = \prod_{i=1}^{\infty} (1 + \frac{1}{x_i}).$

The best solution was submitted by Jang, Kijoung (장기정, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2017-15.

Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 김기택 (수리과학과 2015학번, +3), 김태균 (수리과학과 2016학번, +3), 송교범 (고려대 수학과 2017학번, +3), 어수강 (서울대학교 수학교육과 박사과정, +3), 유찬진 (수리과학과 2015학번, +3), 윤준기 (전기및전자공학부 2014학번, +3), 이본우 (2017학번, +3), 이태영 (수리과학과 2013학번, +3), 조태혁 (수리과학과 2014학번, +3), 최대범 (수리과학과 2016학번, +3), Huy Tung Nguyen (수리과학과 2016학번, +3), 김동률 (수리과학과 2015학번, +2), 이재우 (함양고등학교 2학년, +2).

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Is it possible to color all lattice points ($$\mathbb Z\times \mathbb Z$$) in the plane into two colors such that if four distinct points $$(a,b), (a+c,b), (a,b+d), (a+c,b+d)$$ have the same color, then $$d/c\notin \{1,2,3,4,6\}$$?