Consider any sequence a1,…,an of non-negative integers in {0,1,…,m}. Prove that |{ai+aj+(j−i):1≤i<j≤n}|≥m when m=⌊14n2/3⌋.
The best solution was submitted by 김준홍 (수리과학과 석박통합과정, +4). Congratulations!
Here is the best solution of problem 2025-03.
Another solution was submitted by Anar Rzayev (수리과학과 19학번, +2).
We write tx=(tx0,…,tx5) for x=(x0,…,x5)∈R6 and t∈R. Find all real multivariate polynomials P(x) in x satisfying the following properties:
(a) P(tx)=tdP(x) for all t∈R and x∈R6, where 0≤d≤15 is an integer;
(b) P(x)=0 if xi=xj with i≠j.
Let X and Y be closed manifolds, and suppose X is a finite-sheeted cover of Y. Prove or disprove that if Y has a nontrivial first Betti number, then X also has a nontrivial first Betti number.
The best solution was submitted by Anar Rzayev (수리과학과 19학번, +4). Congratulations!
Here is the best solution of problem 2025-02.
Other solutions were submitted by 김동훈 (수리과학과 22학번, +3), 신민규 (수리과학과 24학번, +3), 성석희 (수리과학과 19학번, +3).
Consider any sequence a1,…,an of non-negative integers in {0,1,…,m}. Prove that |{ai+aj+(j−i):1≤i<j≤n}|≥m when m=⌊14n2/3⌋.
A bonus problem: Can you find a function f(n)=ω(n2/3) such that the above statement is true when m=f(n)? Is there such a function with f(n)=Ω(n)? (You would still get full points without answering the bonus question.)
Find all positive integers a,n such that
1a+1a+1+⋯+1a+n
is an integer.
The best solution was submitted by 박기윤 (전산학부 23학번, +4). Congratulations!
Here is the best solution of problem 2025-01.
Other solutions were submitted by 공기목 (전산학부 20학번, +3), 김동훈 (수리과학과 22학번, +3), 김민서 (수리과학과 19학번, +3), 김준홍 (수리과학과 석박통합과정, +3), 김찬우 (연세대학교 수학과 22학번, +3), 나승균 (수리과학과 23학번, +3), 노희윤 (수리과학과 석박통합과정, +3), 서성욱 (서울대학교 수리과학부 25학번, +3), 신민규 (수리과학과 24학번, +3), 이도엽 (연세대학교 수학과 24학번, +3), 이명규 (전기및전자공학부 20학번, +3), 양준혁 (수리과학과 20학번, +3), 정서윤 (수리과학과 학사과정, +3), 정영훈 (수리과학과 24학번, +3), 채지석 (수리과학과 석박통합과정, +3), 최정담(디지털인문사회과학부 석사과정, +3), 최기범 (한양대학교 졸업생, +3), 최백규 (생명과학과 박사과정, +3). 정지혁 (수리과학과 22학번, +). There were incorrect solutions submitted. Late solutions were not graded.
(Added: The previous best solution has a gap, so we changed the best solution. I apologize for any inconvenience.)
Let X and Y be closed manifolds, and suppose X is a finite-sheeted cover of Y. Prove or disprove that if Y has a nontrivial first Betti number, then X also has a nontrivial first Betti number.
Find all positive integers a,n such that
1a+1a+1+⋯+1a+n
is an integer.
Prove or disprove that every homomorphism π1(X)→π1(X) can be realized as the induced homomorphism of a continuous map X→X.
The best solution was submitted by 김준홍 (KAIST 수리과학과 석박통합과정, +4). Congratulations!
Here is the best solution of problem 2024-21.
Other solutions were submitted by 김찬우 (연세대학교 수학과 22학번, +3), 양준혁 (KAIST 수리과학과 20학번, +3).
Suppose that f:R→R is a continuous function such that the sequence f(x),f(2x),f(3x),… converges to 0 for any x>0. Prove or disprove that lim
The best solution was submitted by 이명규 (KAIST 전산학부 20학번, +4). Congratulations!
Here is the best solution of problem 2024-20.
Other solutions were submitted by 김준홍 (KAIST 수리과학과 석박통합과정, +3), 김찬우 (연세대학교 수학과 22학번, +3), 노희윤 (KAIST 수리과학과 석박통합과정, +3), 양준혁 (KAIST 수리과학과 20학번, +3), 최정담 (KAIST 디지털인문사회과학부 석사과정, +3). There was an incorrect soultion submitted.
Prove or disprove that every homomorphism \pi_1(X) \to \pi_1(X) can be realized as the induced homomorphism of a continuous map X \to X.