Let g(t):[0,+∞)→[0,+∞) be a decreasing continuous function. Assume g(0)=1, and for every s,t≥0 t11g(s+t)≤2024[g(s)]2. Show that g(11)=g(12).
The best solution was submitted by 김준홍 (KAIST 수리과학과 석박통합과정, +4). Congratulations!
Here is the best solution of problem 2024-19.
Other solutions were submitted by 김찬우 (연세대학교 수학과 22학번, +3), 양준혁 (KAIST 수리과학과 20학번, +3), 이명규 (KAIST 전산학부 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
Let f(n) denote the number of possible sequences of length n , where each term is either 0, 1, or -1, such that the product of every three consecutive numbers is nonnegative. Compute f(33).
The best solution was submitted by 신민규 (KAIST 새내기과정학부 24학번, +4). Congratulations!
Here is the best solution of problem 2024-18.
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), 우준서 (KAIST 수리과학과 20학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 채지석 (KAIST 수리과학과 석박통합과정, +3), 최정담 (KAIST 디지털인문사회과학부 석사과정, +3), Daulet Kurmantayev (+3).
Let g(t): [0,+\infty) \to [0,+\infty) be a decreasing continuous function. Assume g(0)=1, and for every s, t \geq 0 t^{11}g(s+t) \leq 2024 \; [g(s)]^2. Show that g(11) = g(12).
Suppose that p(x) is a degree n polynomial with complex coefficients such that p(x) \geq 0 for any real number x . Prove that
p(x) + p'(x) + \dots + p^{(n)}(x) \geq 0
for any real number x .
The best solution was submitted by 이명규 (KAIST 전산학부 20학번, +4). Congratulations!
Here is the best solution of problem 2024-17.
Other solutions were submitted by 김준홍 (KAIST 수리과학과 석박통합과정, +3), 김찬우 (연세대학교 수학과 22학번, +3), 노희윤 (KAIST 수리과학과 석박통합과정, +3), 서성욱 (대전 동산고 3학년, +3), 양준혁 (KAIST 수리과학과 20학번, +3), 최정담 (KAIST 디지털인문사회과학부 석사과정, +3), 최현준 (KAIST 수리과학과 18학번, +3).
Let f(n) denote the number of possible sequences of length n , where each term is either 0, 1, or -1, such that the product of every three consecutive numbers is nonnegative. Compute f(33).
Let A= [a_{ij}]_{1\leq i,j\leq 5} be a 5\times 5 positive definite (real) matrix. Show that the matrix [a_{ij}/(i+j)] is also positive definite.
The best solution was submitted by 김찬우 (연세대학교 수학과 22학번, +4). Congratulations!
Here is the best solution of problem 2024-16.
Other solutions were submitted by 김준홍 (KAIST 수리과학과 석박통합과정, +3), 노희윤 (KAIST 수리과학과 석박통합과정, +3), 서성욱 (대전 동산고 3학년, +3), 신민규 (KAIST 새내기과정학부 24학번, +3), 양준혁 (KAIST 수리과학과 20학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 최정담 (KAIST 디지털인문사회과학부 석사과정, +3).
Suppose that p(x) is a degree n polynomial with complex coefficients such that p(x) \geq 0 for any real number x . Prove that
p(x) + p'(x) + \dots + p^{(n)}(x) \geq 0
for any real number x .
Let A= [a_{ij}]_{1\leq i,j\leq 5} be a 5\times 5 positive definite (real) matrix. Show that the matrix [a_{ij}/(i+j)] is also positive definite.
Is it possible to arrange the numbers 1, 2, 3, \ldots, 2024 in a sequence such that the difference between any two adjacent numbers is greater than 1 but less than 4?
The best solution was submitted by 김준홍 (KAIST 수리과학과 석박통합과정, +4). Congratulations!
Here is the best solution of problem 2024-15.
Other solutions were submitted by 권오관 (연세대학교 수학과 22학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 노희윤 (KAIST 수리과학과 석박통합과정, +3), 서성욱 (대전 동산고 3학년, +3), 신민규 (KAIST 새내기과정학부 24학번, +3), 양준혁 (KAIST 수리과학과 20학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 정영훈 (KAIST 새내기과정학부 24학번, +3), 채지석 (KAIST 수리과학과 석박통합과정, +3), 최백규 (KAIST 생명과학과 박사과정, +3), 최정담 (KAIST 디지털인문사회과학부 석사과정, +3), Anar Rzayev (KAIST 전산학부 19학번, +3), ASKM Sayeef Uddin (KAIST 수리과학과 22학번, +3).