# 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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# Solution: 2021-12 A graduation ceremony

In a graduation ceremony, $$n$$ graduating students form a circle and their diplomas are distributed uniformly at random. Students who have their own diploma leave, and each of the remaining students passes the diploma she has to the student on her right, and this is one round. Again, each student with her own diploma leave and each of the remaining students passes the diploma to the student on her right and repeat this until everyone leaves. What is the probability that this process takes exactly $$k$$ rounds until everyone leaves.

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

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In a graduation ceremony, $$n$$ graduating students form a circle and their diplomas are distributed uniformly at random. Students who have their own diploma leave, and each of the remaining students passes the diploma she has to the student on her right, and this is one round. Again, each student with her own diploma leave and each of the remaining students passes the diploma to the student on her right and repeat this until everyone leaves. What is the probability that this process takes exactly $$k$$ rounds until everyone leaves.

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# Solution: 2021-11 Interesting perfect cubes

Determine if there exist infinitely many perfect cubes such that the sum of the decimal digits coincides with the cube root. If there are only finitely many, how many are there?

The best solution was submitted by 박항 (전산학부 2013학번, +4). Congratulations!

Other solutions were submitted by 강한필 (전산학부 2016학번, +3), 고성훈 (수리과학과 2018학번, +3), 김기수 (수리과학과 2018학번, +3), 최백규 (생명과학과 대학원, +3), 김기택 (2021학번, +3).

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