Is it possible to color all lattice points (Z×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∉{1,2,3,4,6}?
(The next POW problem will be posted on October 20. Happy Chuseok and good luck with your midterm exams.)
Let f(x)∈R[x] be a polynomial of degree at most n such that x2+f(x)2≤1 for all −1≤x≤1. Prove that |f′(x)|≤2(n−1) for all −1≤x≤1.
The best solution was submitted by Huy Tùng Nguyễn (수리과학과 2016학번). Congratulations!
Here is his solution of problem 2017-14.
Alternative solutions were submitted by 유찬진 (수리과학과 2015학번, +3), 이본우 (2017학번, +2). One incorrect solution was submitted.
For x∈(1,2), prove that there exists a unique sequence of positive integers {xi} such that xi+1≥x2i and
x=∞∏i=1(1+1xi).
Let a0=a1=1 and an=nan−1+(n−1)an−2 for n≥2. Find the value of
∞∑n=0(−1)nn!anan+1.
The best solution was submitted by Choi, Daebeom (최대범, 수리과학과 2016학번). Congratulations!
Here is his solution of problem 2017-13.
Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 김동률 (수리과학과 2015학번, +3), 김태균 (수리과학과 2016학번, +3), 유찬진 (수리과학과 2015학번, +3), 윤준기 (전기및전자공학부 2014학번, +3), 이본우 (2017학번, +3), 이태영 (수리과학과 2013학번, +3), 장기정 (수리과학과 2014학번, +3, solution), 조태혁 (수리과학과 2014학번, +3, solution), 최인혁 (물리학과 2015학번, +3), Huy Tung Nguyen (수리과학과 2016학번, +3), 김기택 (수리과학과 2015학번, +2), 이재우 (함양고등학교 2학년, +2), 정의현 (수리과학과 2015학번, +2).
Let f(x)∈R[x] be a polynomial of degree at most n such that x2+f(x)2≤1 for all −1≤x≤1. Prove that |f′(x)|≤2(n−1) for all −1≤x≤1.
Let a0=a1=1 and an=nan−1+(n−1)an−2 for n≥2. Find the value of
∞∑n=0(−1)nn!anan+1.
Let A and B be n×n matrices. Prove that if n is odd and both A+AT and B+BT are invertible, then AB≠0.
The best solution was submitted by Shin, Joonhyung (신준형, 수리과학과 2015학번). Congratulations!
Here is his solution of the problem 2017-12.
Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 김동률 (수리과학과 2015학번, +3), 김태균 (수리과학과 2016학번, +3), 위성군 (수리과학과 2015학번, +3), 유찬진 (수리과학과 2015학번, +3), 윤준기 (전기및전자공학부 2014학번, +3), 이본우 (2017학번, +3), 이준협 (하나고등학교, +3), 이태영 (수리과학과 2013학번, +3), 이형진 (청주대 수학교육과 2011학번, +3), 임성혁 (수리과학과 2016학번, +3), 장기정 (수리과학과 2014학번, +3), 조태혁 (수리과학과 2014학번, +3), 최대범 (수리과학과 2016학번, +3), 최인혁 (물리학과 2015학번, +3), Huy Tung Nguyen (수리과학과 2016학번, +3), Saba Dzmanashvili (2017학번, +3).
Let A and B be n×n matrices. Prove that if n is odd and both A+AT and B+BT are invertible, then AB≠0.