# Notice on 2021-16

POW 2021-16 remains open, as we found gaps in the submitted solutions. Anyone who first submits a correct solution will get the full credit.

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# 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!

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

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# 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 $$a_n$$.

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# Notice on POW 2021-14

POW 2021-14 is still open and anyone who first submits a correct solution will get the full credit.

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# Notice on 2021-14

Please check that POW 2021-14 is slightly changed; there is an additional assumption that $$X \times Y$$ is Hausdorff.

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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!

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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