Find all n≥2 such that the polynomial x^{n}-x^{n-1}-x^{n-2}-…-x-1 is irreducible over the rationals.
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Find all n≥2 such that the polynomial x^{n}-x^{n-1}-x^{n-2}-…-x-1 is irreducible over the rationals.
KAIST POW will take a break for the midterm exam. Good luck to all students!
Next problem will be posted on Oct. 28th.
Let f(x) be a continuous function on I=[a,b], and let g(x) be a differentiable function on I. Let g(a)=0 and c≠0 a constant. Prove that if
|g(x) f(x)+c g′(x)|≤|g(x)| for all x∈I,
then g(x)=0 for all x∈I.
The best solution was submitted by Seungkyun Park (박승균), 수리과학과 2008학번. Congratulations!
Here is his Solution of Problem 2011-18.
Alternative solutions were submitted by 김범수 (수리과학과 2010학번, +3), 장경석 (2011학번, +3), 김태호 (2011학번, +2), 김재훈 (EEWS대학원, +2).
Let f(x) be a continuous function on I=[a,b], and let g(x) be a differentiable function on I. Let g(a)=0 and c≠0 a constant. Prove that if
|g(x) f(x)+c g′(x)|≤|g(x)| for all x∈I,
then g(x)=0 for all x∈I.
Let f(n) be the maximum positive integer m such that the sum of all positive divisors of m is less than or equal to n. Find all positive integers k such that there are infinitely many positive integers n satisfying the equation n-f(n)=k.
The best solution was submitted by Taeho Kim (김태호), 2011학번. Congratulations!
Here is his Solution of Problem 2011-17.
Alternative solutions were submitted by 김범수 (수리과학과 2010학번, +3), 서기원 (수리과학과 2009학번, +3), 박승균 (수리과학과 2008학번, +3), 장경석 (2011학번, +3), 구도완 (해운대고등학교 3학년, +3), 손동현 (유성고등학교 2학년, +2), 어수강 (서울대학교 석사과정, +2).
Update: I forgot to add 최민수 (2011학번, +3) into the list of people submitted alternative solutions.
Let A_{1}, A_{2}, A_{3}, …, A_{n} be finite sets such that |A_{i}| is odd for all 1≤i≤n and |A_{i}∩A_{j}| is even for all 1≤i<j≤n. Prove that it is possible to pick one element a_{i} in each set A_{i} so that a_{1}, a_{2}, …,a_{n} are distinct.
The best solution was submitted by Ilhee Kim (김일희) and Ringi Kim (김린기), Graduate Students, Princeton University.
Here is Solution of Problem 2011-16.
Alternative solutions were submitted by 강동엽 (전산학과 2009학번, +3, solution by 강동엽), 문상혁 & 박상현 (2010학번, +3), 장경석 (2011학번, +3), 이재석 (수리과학과 2007학번, +2). Three incorrect solutions were submitted by B. Kim, J. Lee, Y. Park.
Let f(n) be the maximum positive integer m such that the sum of all positive divisors of m is less than or equal to n. Find all positive integers k such that there are infinitely many positive integers n satisfying the equation n-f(n)=k.