Prove or disprove the following:
For any positive integer \( n \), there exists a polynomial \( P_n \) of degree \( n^2 \) such that
(1) all coefficients of \( P_n \) are integers with absolute value at most \( n^2 \), and
(2) \( 1 \) is a root of \( P_n =0 \) with multiplicity at least \( n \).
For positive integers \(n \geq 2\), let \(a_n = \lceil n/\pi \rceil \) and let \(b_n = \lceil \csc (\pi/n) \rceil \). Is \(a_n = b_n\) for all \(n \neq 3\)?
The best solution was submitted by 김기수 (KAIST 수리과학과 18학번, +4). Congratulations!
Here is the best solution of problem 2022-08.
Other solutions were submitted by 조유리 (문현여고 3학년, +3), 이명규 (KAIST 전산학부 20학번, +3), 박기찬 (KAIST 새내기과정학부 22학번, +3), 여인영 (KAIST 물리학과 20학번, +3).
Let \(A_1,\dots, A_k\) be presidential candidates in a country with \(n \geq 1\) voters with \(k\geq 2\). Candidates themselves are not voters. Each voter has her/his own preference on those \(k\) candidates.
Find maximum \(m\) such that the following scenario is possible where \(A_{k+1}\) indicates the candidate \(A_1\): for each \(i\in [k]\), there are at least \(m\) voters who prefers \(A_i\) to \(A_{i+1}\).
Prove the following identity for \( x, y \in \mathbb{R}^3 \):
\[
\frac{1}{|x-y|} = \frac{1}{\pi^3} \int_{\mathbb{R}^3} \frac{1}{|x-z|^2} \frac{1}{|y-z|^2} dz.
\]
The best solution was submitted by 조유리 (문현여고 3학년, +4). Congratulations!
Here is the best solution of problem 2022-07.
Other solutions were submitted by 이종민 (KAIST 물리학과 21학번, +3), 박기찬 (KAIST 새내기과정학부 22학번, +3), 김기수 (KAIST 수리과학과 18학번, +3), 여인영 (KAIST 물리학과 20학번, +3), 채지석 (KAIST 수리과학과 대학원생, +3), 이상민 (KAIST 수리과학과 대학원생, +3), 나영준 (연세대학교 의학과 18학번, +3).
For positive integers \(n \geq 2\), let \(a_n = \lceil n/\pi \rceil \) and let \(b_n = \lceil \csc (\pi/n) \rceil \). Is \(a_n = b_n\) for all \(n \neq 3\)?
Solutions are due May 13th (Friday), 6PM, and it will remain open if nobody solved it.
Prove the following identity for \( x, y \in \mathbb{R}^3 \):
\[
\frac{1}{|x-y|} = \frac{1}{\pi^3} \int_{\mathbb{R}^3} \frac{1}{|x-z|^2} \frac{1}{|y-z|^2} dz.
\]
Solutions are due May 6th (Friday), 6PM, and it will remain open if nobody solved it.
We have an expression \(x_0 \div x_1 \div x_2 \div \dots \div x_n\). A way of putting \(n-1\) left parentheses and \(n-1\) right parenthese is called ‘parenthesization’ if it is valid and completely clarify the order of operations. For example, when \(n=3\), we have the following five parenthesizations.
\[ ((x_0\div x_1)\div x_2)\div x_3, \enspace (x_0\div (x_1\div x_2))\div x_3, \enspace (x_0\div x_1)\div (x_2\div x_3),\]
\[x_0\div ((x_1\div x_2)\div x_3), \enspace x_0\div (x_1\div (x_2\div x_3)).
\]
(a) For an integer \(n\), how many parenthesizations are there?
(b) For each parenthesization, we can compute the expression to obtain a fraction with some variables in the numerator and some variable in the denominator. For an integer \(n\), determine which fraction occur most often. How many times does it occur?
The best solution was submitted by 나영준 (연세대학교 의학과 18학번, +4). Congratulations!
Here is the best solution of problem 2022-06.
Other (incomplete) solutions were submitted by 조유리 (문현여고 3학년, +2), 이명규 (KAIST 전산학부 20학번, +2), 박기찬 (KAIST 새내기과정학부 22학번, +2), Antonio Recuero Buleje (+2).
POW will resume on Apr. 29.
Show that there do not exist perfect squares a, b, c such that \(a^2 + b^2 = c^2\), provided that a, b, c are nonzero integers.
The best solution was submitted by 박준성 (KAIST 수리과학과 19학번, +4). Congratulations!
Here is the best solution of problem 2022-05.
Other solutions were submitted by 조유리 (문현여고 3학년, +3), 김예곤 (KAIST 수리과학과 19학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 문강연 (KAIST 새내기과정학부 22학번, +3), 박기찬 (KAIST 새내기과정학부 22학번, +3), 이호빈 (KAIST 수리과학과 대학원생, +3), 김기수 (KAIST 수리과학과 18학번, +3), 이재욱 (KAIST 전기및전자공학부 대학원생, +3).
We have an expression \(x_0 \div x_1 \div x_2 \div \dots \div x_n\). A way of putting \(n-1\) left parentheses and \(n-1\) right parenthese is called ‘parenthesization’ if it is valid and completely clarify the order of operations. For example, when \(n=3\), we have the following five parenthesizations.
\[ ((x_0\div x_1)\div x_2)\div x_3, \enspace (x_0\div (x_1\div x_2))\div x_3, \enspace (x_0\div x_1)\div (x_2\div x_3),\]
\[x_0\div ((x_1\div x_2)\div x_3), \enspace x_0\div (x_1\div (x_2\div x_3)).
\]
(a) For an integer \(n\), how many parenthesizations are there?
(b) For each parenthesization, we can compute the expression to obtain a fraction with some variables in the numerator and some variable in the denominator. For an integer \(n\), determine which fraction occur most often. How many times does it occur?