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.
Solution: 2014-03 Subadditive function
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.)
2014-03 Subadditive function
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
\[ f(n_1) + f(n_2) + \cdots + f(n_M) \geq 0. \]
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.).
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.
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.
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 \( n \to \infty \).
Math Problem of the Week 2014 will begin at March 7.
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.
Concluding Fall 2013
This semester, we have several ties including 3 perfect scorers. The top 5 participants of the semester are:
- T-1st: 박민재 (11학번): 35 points
- T-1st: 진우영 (12학번): 35 points
- T-1rd: 정성진 (13학번): 35 points
- T-4th: 김호진 (09학번): 25 points
- T-4th: 박훈민 (13학번): 25 points
Hearty congratulations to the prize winners!
We thank all of the participants for the nice solutions and your interest you showed for POW. We hope to see you next semester with better problems.
Solution: 2013-23 Polynomials with rational zeros
Find all polynomials \( P(x) = a_n x^n + \cdots + a_1 x + a_0 \) satisfying (i) \( a_n \neq 0 \), (ii) \( (a_0, a_1, \cdots, a_n) \) is a permutation of \( (0, 1, \cdots, n) \), and (iii) all zeros of \( P(x) \) are rational.
The best solution was submitted by 전한솔. Congratulations!
Similar solutions are submitted by 김호진(+3), 박민재(+3), 엄태현(+3), 정성진(+3), 진우영(+3), 한대진(+3). Thank you for your participation.