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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