# 2023-18 Degrees of a graph

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.

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# Solution: 2023-17 Comparing area of triangles

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!

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).

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# 2023-17 Comparing area of triangles

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 $$T \leq 3A/2.$$

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# Solution: 2023-16 Zeros in a sequence

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!

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).

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