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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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.

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.)

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

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

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.).

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.

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.

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 first problem of 2014 spring semester will be posted at March 7. As usual, problems will be posted on every Friday at 3PM and solutions will be due next Wednesday at noon. Please submit your solution to * pow@mathsci.kaist.ac.kr* by email.

In this semester, POW will be co-organized by Prof. Sang-il Oum and Prof. Ji Oon Lee. Thus, we’ve got a new email address for POW.