Say a natural number \(n\) is a cyclically perfect if one can arrange the numbers from 1 to \(n\) on the circle without a repeat so that the sum of any two consecutive numbers is a perfect square. Show that 32 is the smallest cyclically perfect number. Find the second smallest cyclically perfect number.

The best solution was submitted by 전해구 (기계공학과 졸업생, +4). Congratulations!

Here is the best solution of problem 2021-20.

Suppose that \( a_1 + a_2 + \dots + a_n =0 \) for real numbers \( a_1, a_2, \dots, a_n \) and \( n \geq 2\). Set \( a_{n+i}=a_i \) for \( i=1, 2, \dots \). Prove that
\[
\sum_{i=1}^n \frac{1}{a_i (a_i+a_{i+1}) (a_i+a_{i+1}+a_{i+2}) \dots (a_i+a_{i+1}+\dots+a_{i+n-2})} =0
\]
if the denominators are nonzero.

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

Here is the best solution of problem 2021-19.

Let \(T\) be a tree (an acyclic connected graph) on the vertex set \([n]=\{1,\dots, n\}\).
Let \(A\) be the adjacency matrix of \(T\), i.e., the \(n\times n\) matrix with \(A_{ij} = 1\) if \(i\) and \(j\) are adjacent in \(T\) and \(A_{ij}=0\) otherwise. Prove that the number of nonnegative eigenvalues of \(A\) equals to the size of the largest independent set of \(T\). Here, an independent set is a set of vertices where no two vertices in the set are adjacent.

The best solution was submitted by 전해구 (기계공학과 졸업생, +4). Congratulations!

Here is the best solution of problem 2021-18.

For a given positive integer \( n \) and a real number \( a \), find the maximum constant \( b \) such that
\[
x_1^n + x_2^n + \dots + x_n^n + a x_1 x_2 \dots x_n \geq b (x_1 + x_2 + \dots + x_n)^n
\]
for any non-negative \( x_1, x_2, \dots, x_n \).

The best solution was submitted by 전해구 (기계공학과 졸업생, +4). Congratulations!

Here is the best solution of problem 2021-16.

Prove or disprove the following:

There exist an infinite sequence of functions \( f_n: [0, 1] \to \mathbb{R} , n=1, 2, \dots \) ) such that

(1) \( f_n(0) = f_n(1) = 0 \) for any \( n \),

(2) \( f_n(\frac{a+b}{2}) \leq f_n(a) + f_n(b) \) for any \( a, b \in [0, 1] \),

(3) \( f_n – c f_m \) is not identically zero for any \( c \in \mathbb{R} \) and \( n \neq m \).

The best solution was submitted by 김기택 (수리과학과 대학원생, +4). Congratulations!

Here is the best solution of problem 2021-13.

Other solutions were submitted by 강한필 (전산학부 2016학번, +3), 고성훈 (수리과학과 2018학번, +3), 김민서 (수리과학과 2019학번, +3), 박정우 (수리과학과 2019학번, +3), 신주홍 (수리과학과 2020학번, +3), 이도현 (수리과학과 2018학번, +3), 이본우 (수리과학과 2017학번, +3), 이호빈 (수리과학과 대학원생, +3).

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!

Here is the best solution of problem 2021-12.

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!

Here is the best solution of problem 2021-11.

Other solutions were submitted by 강한필 (전산학부 2016학번, +3), 고성훈 (수리과학과 2018학번, +3), 김기수 (수리과학과 2018학번, +3), 최백규 (생명과학과 대학원, +3), 김기택 (2021학번, +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
\[
\int_0^1 [f(t)]^2 dt \geq \int_0^1 t f(t) dt.
\]

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).