Show that for any triangle T and any Jordan curve C in the Euclidean plane, there exists a triangle inscribed in C which is similar to T.
The best solution was submitted by 강한필 (전산학부 2016학번, +4). Congratulations!
Here is his solution of problem 2021-02.
Other solutions was submitted by 하석민 (수리과학과 2017학번, +3), 박은아 (수리과학과 2015학번, +2).
Prove that for any given positive integer \( n \), there exists a sequence of the following operations that transforms \( n \) to a single-digit number (in decimal representation).
1) multiply a given positive integer by any positive integer.
2) remove all zeros in the decimal representation of a given positive integer.
The best solution was submitted by 강한필 (전산학부 2016학번, +4). Congratulations!
Here is his solution of problem 2021-01.
Other solutions was submitted by 김기수 (수리과학과 2018학번), 박은아 (수리과학과 2015학번, +3), 전해구 (기계공학과 졸업생).
Show that for any triangle T and any Jordan curve C in the Euclidean plane, there exists a triangle inscribed in C which is similar to T.
Prove that for any given positive integer \( n \), there exists a sequence of the following operations that transforms \( n \) to a single-digit number (in decimal representation).
1) multiply a given positive integer by any positive integer.
2) remove all zeros in the decimal representation of a given positive integer.
For each \( i \in \mathbb{N}\), let \(F_i\) be the \(i\)-th Fibonacci number where \(F_0=0, F_1=1\) and \(F_{i+1}=F_{i}+F_{i-1}\) for each \(i\geq 1\).
For \(n>m\), we divide \(F_n\) by \(F_m\) to obtain the remainder \(R\). Prove that either \(R\) or \(F_m-R\) is a Fibonacci number.
The best solution was submitted by 고성훈 (수리과학과 2018학번, +4). Congratulations!
Here is his solution of problem 2020-24.
Other solutions was submitted by Abdirakhman Ismail (2020학번), 이준호 (수리과학과 2016학번, +3).
For each \( i \in \mathbb{N}\), let \(F_i\) be the \(i\)-th Fibonacci number where \(F_0=0, F_1=1\) and \(F_{i+1}=F_{i}+F_{i-1}\) for each \(i\geq 1\).
For \(n>m\), we divide \(F_n\) by \(F_m\) to obtain the remainder \(R\). Prove that either \(R\) or \(F_m-R\) is a Fibonacci number.
Suppose we choose a point on the unit circle in the plane at random with the uniform probability measure on the circle. When we choose n points in that way, what is the probability of the n-gon obtained as the convex hull of the chosen points has the area bigger than \( \pi/2 \) in terms of n?
POW 2020-21 is still open and anyone who first submits a correct solution will get the full credit.
Let \( S \) be the unit sphere in \( \mathbb{R}^n \), centered at the origin, and \( P_1 P_2 \dots P_{n+1} \) a regular simplex inscribed in \( S \). Prove that for a point \( P \) inside \( S \),
\[
\sum_{i=1}^{n+1} (PP_i)^4
\]
depends only on the distance \( OP \) (and \(n\)).
The best solution was submitted by 이준호 (수리과학과 2016학번, +4). Congratulations!
Here is his solution of problem 2020-22.
Other solutions was submitted by 고성훈 (수리과학과 2018학번, +3), 채지석 (수리과학과 2016학번, +3).
Let \( S \) be the unit sphere in \( \mathbb{R}^n \), centered at the origin, and \( P_1 P_2 \dots P_{n+1} \) a regular simplex inscribed in \( S \). Prove that for a point \( P \) inside \( S \),
\[
\sum_{i=1}^{n+1} (PP_i)^4
\]
depends only on the distance \( OP \) (and \(n\)).