# 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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# 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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# Solution: 2021-10 Integral inequality

Let $$f: [0, 1] \to \mathbb{R}$$ be a continuous function satisfying
$\int_x^1 f(t) dt \geq \int_x^1 t\, dt$
for all $$x \in [0, 1]$$. Prove that
$\int_0^1 [f(t)]^2 dt \geq \int_0^1 t f(t) dt.$

The best solution was submitted by 김기택 (2021학번, +4). Congratulations!

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

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# Solution: 2021-09 Monochromatic solution of an equation

For given $$k\in \mathbb{N}$$, determine the minimum natural number $$n$$ satisfying the following: no matter how one colors each number in $$\{1,2,\dots, n\}$$ red or blue, there always exists (not necessarily distinct) numbers $$x_0, x_1,\dots, x_k \in [n]$$ with the same color satisfying $$x_1+\dots + x_k = x_0$$.

The best solution was submitted by an anonymous participant. Congratulations!

Here is his/her solution of problem 2021-09.

Other solutions were submitted by 고성훈 (수리과학과 2018학번, +3), 김기수 (수리과학과 2018학번, +3).

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# Solution: 2021-08 Self-antipodal sets on the sphere

Prove or disprove that if C is any nonempty connected, closed, self-antipodal (ie., invariant under the antipodal map) set on $$S^2$$, then it equals the zero locus of an odd, smooth function $$f:S^2 -> \mathbb{R}$$.

The best solution was submitted by 신준형 (수리과학과 2015학번, +4). Congratulations!

Here is his solution of problem 2021-08.

Another solution was submitted by 고성훈 (수리과학과 2018학번, +2).

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# Solution: 2021-07 Odd determinant

Let $$A_N$$ be an $$N \times N$$ matrix whose entries are i.i.d. Bernoulli random variables with probability $$1/2$$, i.e.,

$\mathbb{P}( (A_N)_{ij} =0) = \mathbb{P}( (A_N)_{ij} =1) = \frac{1}{2}.$

Let $$p_N$$ be the probability that $$\det A_N$$ is odd. Find $$\lim_{N \to \infty} p_N$$.

The best solution was submitted by 강한필 (전산학부 2016학번, +4). Congratulations!

Here is his solution of problem 2021-07.

Another solution was submitted by 고성훈 (수리과학과 2018학번, +3).

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# Solution: 2021-06 A nondecreasing subsequence

Let $$\mathcal{A}_n$$ be the collection of all sequences $$\mathbf{a}= (a_1,\dots, a_n)$$ with $$a_i \in [i]$$ for all $$i\in [n]=\{1,2,\dots, n\}$$. A nondecreasing $$k$$-subsequence of $$\mathbf{a}$$ is a subsequence $$(a_{i_1}, a_{i_2},\dots, a_{i_k})$$ such that $$i_1< i_2< \dots < i_k$$ and $$a_{i_1}\leq a_{i_2}\leq \dots \leq a_{i_k}$$. For given $$k$$, determine the smallest $$n$$ such that any sequence $$\mathbf{a}\in \mathcal{A}_n$$ has a nondecreasing $$k$$-subsequence.

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

Here is his solution of problem 2021-06.

Another solution was submitted by 강한필 (전산학부 2016학번, +3).

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# Solution: 2021-05 Finite generation of a group

Prove or disprove that if all elements of an infinite group G has order less than n for some positive integer n, then G is finitely generated.

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

Here is his solution of problem 2021-05.

Other solutions were submitted by 강한필 (전산학부 2016학번, +3), 고성훈 (수리과학과 2018학번, +3), Late solutions are not graded.

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# Solution: 2021-04 Product of matrices

For an $$n \times n$$ matrix $$M$$ with real eigenvalues, let $$\lambda(M)$$ be the largest eigenvalue of $$M$$. Prove that for any positive integer $$r$$ and positive semidefinite matrices $$A, B$$,

$[\lambda(A^m B^m)]^{1/m} \leq [\lambda(A^{m+1} B^{m+1})]^{1/(m+1)}.$

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

Here is his solution of problem 2021-04.

Another solutions was submitted by 김건우 (수리과학과 2017학번, +3),

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