Monthly Archives: September 2021

2021-16 Optimal constant

For a given positive integer \( n \) and a real number \( a \), find the maximum constant \( b \) such that
\[
x_1^n + x_2^n + \dots + x_n^n + a x_1 x_2 \dots x_n \geq b (x_1 + x_2 + \dots + x_n)^n
\]
for any non-negative \( x_1, x_2, \dots, x_n \).

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Solution: 2021-15 Triangles with integer side lengths

For a natural number \(n\), let \(a_n\) be the number of congruence classes of triangles whose all three sides have integer length and its perimeter is \(n\). Obtain a formula for \(a_n\).

The best solution was submitted by 이도현 (수리과학과 2018학번, +4). Congratulations!

Here is the best solution of problem 2021-15.

Other solutions were submitted by 강한필 (전산학부 2016학번, +3), 고성훈 (수리과학과 2018학번, +3), 전해구 (기계공학과 졸업생, +3).

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2021-14 Perfectly normal product

Let X, Y be compact spaces. Suppose \(X \times Y\) is perfectly normal, i.e, for every disjoint closed subsets E, F in \(X \times Y\), there exists a continuous function \( f: X \times Y \to [0, 1] \subset \mathbb{R} \) such that \( f^{-1}(0) = E, f^{-1}(1) = F \). Is it true that at least one of X and Y is metrizable?

(added Sep. 11, 8AM: Assume further that \( X \times Y\) is Hausdorff.)

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Solution: 2021-13 Not convex

Prove or disprove the following:

There exist an infinite sequence of functions \( f_n: [0, 1] \to \mathbb{R} , n=1, 2, \dots \) ) such that

(1) \( f_n(0) = f_n(1) = 0 \) for any \( n \),

(2) \( f_n(\frac{a+b}{2}) \leq f_n(a) + f_n(b) \) for any \( a, b \in [0, 1] \),

(3) \( f_n – c f_m \) is not identically zero for any \( c \in \mathbb{R} \) and \( n \neq m \).

The best solution was submitted by 김기택 (수리과학과 대학원생, +4). Congratulations!

Here is the best solution of problem 2021-13.

Other solutions were submitted by 강한필 (전산학부 2016학번, +3), 고성훈 (수리과학과 2018학번, +3), 김민서 (수리과학과 2019학번, +3), 박정우 (수리과학과 2019학번, +3), 신주홍 (수리과학과 2020학번, +3), 이도현 (수리과학과 2018학번, +3), 이본우 (수리과학과 2017학번, +3), 이호빈 (수리과학과 대학원생, +3).

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2021-13 Not convex

Prove or disprove the following:

There exist an infinite sequence of functions \( f_n: [0, 1] \to \mathbb{R} , n=1, 2, \dots \) ) such that

(1) ( f_n(0) = f_n(1) = 0 ) for any ( n ),

(2) ( f_n(\frac{a+b}{2}) \leq f_n(a) + f_n(b) ) for any ( a, b \in [0, 1] ),

(3) ( f_n – c f_m ) is not identically zero for any ( c \in \mathbb{R} ) and ( n \neq m ).

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