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
\[
\int_0^1 [f(t)]^2 dt \geq \int_0^1 t f(t) dt.
\]

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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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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 \(x_1+\dots + x_k = x_0\).

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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 \(f:S^2 -> \mathbb{R}\).  

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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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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 \( \lim_{N \to \infty} p_N \).

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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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