# 2011-19 Irreducible polynomial

Find all n≥2 such that the polynomial xn-xn-1-xn-2-…-x-1 is irreducible over the rationals.

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# Break for the midterm exam

KAIST POW will take a break for the midterm exam. Good luck to all students!

Next problem will be posted on Oct. 28th.

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# Solution: 2011-18 Continuous Function and Differentiable Function

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(xf(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).

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# 2011-18 Continuous Function and Differentiable Function

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(xf(x)+c g′(x)|≤|g(x)| for all x∈I,

then g(x)=0 for all x∈I.

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# Solution: 2011-17 Infinitely many solutions

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.

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# Solution: 2011-16 Odd sets with Even Intersection

Let A1, A2, A3, …, An be finite sets such that |Ai| is odd for all 1≤i≤n and |Ai∩Aj| is even for all 1≤i<j≤n. Prove that it is possible to pick one element ai in each set Ai so that a1, a2, …,an 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.

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