# 2014-04 Integer pairs

Prove that there exist infinitely many pairs of positive integers $$(m, n)$$ satisfying the following properties:

(1) gcd$$(m, n) = 1$$.

(2) $$(x+m)^3 = nx$$ has three distinct integer solutions.

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Let $$f: [0, \infty) \to \mathbb{R}$$ be a function satisfying the following conditions:

(1) For any $$x, y \geq 0$$, $$f(x+y) \geq f(x)+f(y)$$.

(2) For any $$x \in [0, 2]$$, $$f(x) \geq x^2 – x$$.

Prove that, for any positive integer $$M$$ and positive reals $$n_1, n_2, \cdots, n_M$$ with $$n_1 + n_2 + \cdots + n_M = M$$, we have

$f(n_1) + f(n_2) + \cdots + f(n_M) \geq 0.$

The best solution was submitted by 박훈민. Congratulations!

Alternative solutions were submitted by 김경석 (+3), 김일희 (+3), 안현수 (+3), 어수강 (+3), 윤성철 (+3), 이영민 (+3), 이종원 (+3), 장경석 (+3), 장기정 (+3), 정성진 (+3), 정진야 (+3), 조준영 (+3), 채석주 (+3), 황성호 (+3). Incorrect solutions were submitted by K.W.J., K.H.S., N.J.H, M.K.Y., S.W.C., L.H.B., C.W.H. (Some initials here might have been improperly chosen.)

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Let $$f: [0, \infty) \to \mathbb{R}$$ be a function satisfying the following conditions:

(1) For any $$x, y \geq 0$$, $$f(x+y) \geq f(x)+f(y)$$.

(2) For any $$x \in [0, 2]$$, $$f(x) \geq x^2 – x$$.

Prove that, for any positive integer $$M$$ and positive reals $$n_1, n_2, \cdots, n_M$$ with $$n_1 + n_2 + \cdots + n_M = M$$, we have

$f(n_1) + f(n_2) + \cdots + f(n_M) \geq 0.$

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# Solution: 2014-02 Series

Determine all positive integers $$\ell$$ such that $\sum_{n=1}^\infty \frac{n^3}{(n+1)(n+2)(n+3)\cdots (n+\ell)}$ converges and if it converges, then compute its value.

The best solution was submitted by 황성호 (2013학번). Congratulations!

Alternative solutions were submitted by 박훈민 (+3), 이종원 (+3), 채석주 (+3), 이영민 (+2), 조준영 (+2),정성진 (+3), 장기정 (+3), 오동우 (+3), 이상철 (+3), 어수강 (+3), 엄문용 (+3), 윤성철 (+3), 전한울 (+3), 박경호 (+2), 한대진 (+2), 서진솔 (+2), 이시우 (+2). Four incorrect solutions were submitted (J.K.S., N.J.H., A.H.S., C.J.H.).

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# 2014-02 Series

Determine all positive integers $$\ell$$ such that $\sum_{n=1}^\infty \frac{n^3}{(n+1)(n+2)(n+3)\cdots (n+\ell)}$ converges and if it converges, then compute its value.

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# Solution: 2014-01 Uniform convergence

Let $$f$$ be a real-valued continuous function on $$[ 0, 1]$$. For a positive integer $$n$$, define
$B_n(f; x) = \sum_{j=0}^n f( \frac{j}{n}) {n \choose j} x^j (1-x)^{n-j}.$
Prove that $$B_n (f; x)$$ converges to $$f$$ uniformly on $$[0, 1 ]$$ as $$n \to \infty$$.

The best solution was submitted by 김범수. Congratulations!

Similar solutions are submitted by 권현우(+3), 박경호(+3), 오동우(+3), 이시우(+3), 이종원(+3), 이주호(+3), 장경석(+3), 장기정(+3), 정성진(+3), 정진야(+3), 조준영(+3), 채석주(+3), 한대진(+3), 황성호(+3). Thank you for your participation.

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# 2014-01 Uniform convergence

Let $$f$$ be a real-valued continuous function on $$[ 0, 1]$$. For a positive integer $$n$$, define
$B_n(f; x) = \sum_{j=0}^n f( \frac{j}{n}) {n \choose j} x^j (1-x)^{n-j}.$
Prove that $$B_n (f; x)$$ converges to $$f$$ uniformly on $$[0, 1 ]$$ as $$n \to \infty$$.

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