Find the following limit:
\[
\lim_{n \to \infty} \left( \frac{\sum_{k=1}^{n+2} k^k}{\sum_{k=1}^{n+1} k^k} – \frac{\sum_{k=1}^{n+1} k^k}{\sum_{k=1}^{n} k^k} \right)
\]
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Find the following limit:
\[
\lim_{n \to \infty} \left( \frac{\sum_{k=1}^{n+2} k^k}{\sum_{k=1}^{n+1} k^k} – \frac{\sum_{k=1}^{n+1} k^k}{\sum_{k=1}^{n} k^k} \right)
\]
Let \( N \) be the number of ordered tuples of positive integers \( (a_1, a_2, \dots, a_{27}) \) such that \( \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_{27}} = 1\). Compute the remainder of \( N \) when \( N \) is divided by \(33 \).
The best solution was submitted by 이명규 (KAIST 전산학부 20학번, +4). Congratulations!
Here is the best solution of problem 2023-19.
Other solutions were submitted by 강지민 (세마고 3학년, +3), 김기수 (KAIST 수리과학과 18학번, +3), 김민서 (KAIST 수리과학과 19학번, +3), 김준홍 (KAIST 수리과학과 20학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 이도현 (KAIST 수리과학과 석박통합과정 23학번, +3), 조현준 (KAIST 수리과학과 22학번, +3), 지은성 (KAIST 수리과학과 20학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), Adnan Sadik (KAIST 새내기과정학부 23학번, +3), Dzhamalov Omurbek (KAIST 전산학부 22학번, +3), Kharchenka Yuliya (KAIST 물리학과 22학번, +3), Muhammadfiruz Hasanov (+3), Aiden Stock (+3).
Can we find a sequence \(a_i, i=0,1,2,…\) with the following property: for each given integer \(n\geq 0\), we have \[\lim_{L\to +\infty}\sum_{i=0}^L 2^{ni} |a_i|\leq 23^{(n+11)^{10}} \quad \text{ and }\quad \lim_{L\to +\infty}\sum_{i=0}^L 2^{ni} a_i = (-1)^n ?\]
Find all integers \( n \geq 8 \) such that there exists a simple graph with \( n \) vertices whose degrees are as follows:
(i) \( (n-4) \) vertices of the graph are with degrees \( 4, 5, 6, \dots, n-2, n-1 \), respectively.
(ii) The other \( 4 \) vertices are with degrees \( n-2, n-2, n-1, n-1 \), respectively.
The best solution was submitted by 이도현 (KAIST 수리과학과 석박통합과정 23학번, +4). Congratulations!
Here is the best solution of problem 2023-18.
Other solutions were submitted by 강지민 (세마고 3학년, +3), 김기수 (KAIST 수리과학과 18학번, +3), 김민서 (KAIST 수리과학과 19학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 나경민 (KAIST 전산학부 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 전해구 (KAIST 기계공학과 졸업생, +3), 조현준 (KAIST 수리과학과 22학번, +3), 지은성 (KAIST 수리과학과 20학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), 최민규 (한양대학교 의과대학 졸업생, +3), Adnan Sadik (KAIST 새내기과정학부 23학번, +3), Dzhamalov Omurbek (KAIST 전산학부 22학번, +3), Muhammadfiruz Hasanov (+3).
In POW 2013-19, there was a typo in the assumption on \( a_1, \dots, a_{27} \). The inequality is now corrected to the equality.
Let \(N\) be the number of ordered tuples of positive integers \( (a_1,a_2,\ldots, a_{27} )\) such that \( \frac{1}{a_1} + \frac{1}{a_2} + \cdots +\frac{1}{a_{27}} = 1\). Compute the remainder of \(N\) when \(N\) is divided by \(3\).
Find all integers \( n \geq 8 \) such that there exists a simple graph with \( n \) vertices whose degrees are as follows:
(i) \( (n-4) \) vertices of the graph are with degrees \( 4, 5, 6, \dots, n-2, n-1 \), respectively.
(ii) The other \( 4 \) vertices are with degrees \( n-2, n-2, n-1, n-1 \), respectively.
Let \(f(x) = x^4 + (2-a)x^3 – (2a+1)x^2 + (a-2)x + 2a\) for some \(a \geq 2\). Draw two tangent lines of its graph at the point \((-1,0)\) and \((1,0)\) and let \(P\) be the intersection point. Denote by \(T\) the area of the triangle whose vertices are \((-1,0), (1,0)\) and \(P\). Let \(A\) be the area of domain enclosed by the interval \([-1,1]\) and the graph of the function on this interval. Show that \(T \leq 3A/2.\)
The best solution was submitted by 서성욱(동산고 2학년, +4). Congratulations!
Here is the best solution of problem 2023-17.
Other solutions were submitted by 강지민 (세마고 3학년, +3), 김민서 (KAIST 수리과학과 19학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 여인영 (KAIST 물리학과 20학번, +3), 이도현 (KAIST 수리과학과 석박통합과정 23학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 조현준 (KAIST 수리과학과 22학번, +3), 지은성 (KAIST 수리과학과 20학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), 최민규 (한양대학교 의과대학 졸업생, +3), Adnan Sadik (KAIST 새내기과정학부 23학번, +3), Muhammadfiruz Hasanov (+3).
Let \(f(x) = x^4 + (2-a)x^3 – (2a+1)x^2 + (a-2)x + 2a\) for some \(a \geq 2\). Draw two tangent lines of its graph at the point \((-1,0)\) and \((1,0)\) and let \(P\) be the intersection point. Denote by \(T\) the area of the triangle whose vertices are \((-1,0), (1,0)\) and \(P\). Let \(A\) be the area of domain enclosed by the interval \([-1,1]\) and the graph of the function on this interval. Show that \(T \leq 3A/2.\)
Define the sequence \( x_n \) by \( x_1 = 0 \) and
\[
x_n = x_{\lfloor n/2 \rfloor} + (-1)^{n(n+1)/2}
\]
for \( n \geq 2\). Find the number of \( n \leq 2023 \) such that \( x_n = 0 \).
The best solution was submitted by 김찬우 (연세대학교 수학과 22학번, +4). Congratulations!
Here is the best solution of problem 2023-16.
Other solutions were submitted by 김기수 (KAIST 수리과학과 18학번, +3), 김민서 (KAIST 수리과학과 19학번, +3), 김준홍 (KAIST 수리과학과 20학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 서성욱 (동산고 2학년, +3), 여인영 (KAIST 물리학과 20학번, +3),이도현 (KAIST 수리과학과 석박통합과정 23학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 전해규 (KAIST 기계공학과 졸업생, +3), 조현준 (KAIST 수리과학과 22학번, +3), 최백규 (KAIST 생명과학과 박사과정 20학번, +3), Adnan Sadik (KAIST 새내기과정학부 23학번, +3), Muhammadfiruz Hasanov (+3), 강지민 (세마고 3학년, +2), 지은성 (KAIST 수리과학과 20학번, +2), 최민규 (한양대학교 의과대학 졸업생, +2), Eun Chan (+2).