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?

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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?

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!

Here is the best solution of problem 2021-10.

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

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).

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.

\]

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).

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

POW 2021-08 is still open and anyone who first submits a correct solution will get the full credit.

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 \(f:S^2 -> \mathbb{R}\).

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).