Solution: 2023-14 Dividing polynomials

Let \(f(t)=(t^{pq}-1)(t-1) \) and \(g(t)=(t^{p}-1)(t^q-1) \) where \(p\) and \(q\) are relatively prime positive integers. Prove that \(\frac{f(t)}{g(t)}\) can be written as a polynomial where it has just \(1\) or \(-1\) as coefficients. (For example, when \(p=2\) and \(q=3\), we have that \(\frac{f(t)}{g(t)} = t^2-t+1\).)

The best solution was submitted by 김준홍 (KAIST 수리과학과 20학번, +4). Congratulations!

Here is the best solution of problem 2023-14.

Other solutions were submitted by 강지민 (세마고 3학년, +3), 김기수 (KAIST 수리과학과 18학번, +3), 김민서 (KAIST 수리과학과 19학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 이도현 (KAIST 수리과학과 석박통합과정 23학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 전해구 (KAIST 기계공학과 졸업생, +3), 조현준 (KAIST 수리과학과 22학번, +4), 지은성 (KAIST 수리과학과 20학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +3). Muhammadfiruz Hasanov (+3).

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2023-15 An inequality for complex polynomials

Let \(p(z), q(z) \)and \(r(z)\) be polynomials with complex coefficients in the complex plane. Suppose that \(|p(z)| + |q(z)| \leq |r(z)|\) for every \(z\). Show that there exist two complex numbers \( a,b \) such that \(|a|^2 +|b|^2 =1\) and \( a p(z) + bq(z) =0 \) for every \(z\).
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Solution: 2023-13 Range of a function on subintervals

Prove or disprove the existence of a function \( f:[0, 1] \to [0, 1] \) with the following property:

for any interval \( (a, b) \subset [0, 1] \) with \( a<b \), \( f((a, b)) = [0, 1] \).

The best solution was submitted by 조현준 (KAIST 수리과학과 22학번, +4). Congratulations!

Here is the best solution of problem 2023-13.

Other solutions were submitted by 김기수 (KAIST 수리과학과 18학번, +3), 김민서 (KAIST 수리과학과 19학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 박준성 (KAIST 수리과학과 석박통합과정 22학번, +3), 오동언 (서울대학교 의과대학 19하번, +3), 이도현 (KAIST 수리과학과 석박통합과정 23학번, +3), 지은성 (KAIST 수리과학과 20학번, +3), Eun U (+3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +2), 이명규 (KAIST 전산학부 20학번, +2), Anar Rzayev (KAIST 전산학부 19학번, +2). There were incorrect solutions.

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2023-14 Dividing polynomials

Let \(f(t)=(t^{pq}-1)(t-1) \) and \(g(t)=(t^{p}-1)(t^q-1) \) where \(p\) and \(q\) are relatively prime positive integers. Prove that \(\frac{f(t)}{g(t)}\) can be written as a polynomial where it has just \(1\) or \(-1\) as coefficients. (For example, when \(p=2\) and \(q=3\), we have that \(\frac{f(t)}{g(t)} = t^2-t+1\).)

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Solution: 2023-12 Pairs promoting diversity

Let \(p\) be a prime number at least three and let \(k\) be a positive integer smaller than \(p\). Given \(a_1,\dots, a_k\in \mathbb{F}_p\) and distinct elements \(b_1,\dots, b_k\in \mathbb{F}_p\), prove that there exists a permutation \(\sigma\) of \([k]\) such that the values of \(a_i + b_{\sigma(i)}\) are distinct modulo \(p\).

The best solution was submitted by 이명규 (KAIST 전산학부, +4). Congratulations!

Here is the best solution of problem 2023-12.

Other solutions were submitted by 김찬우 (연세대학교 수학과 22학번, +3), 박준성 (KAIST 수리과학과 석박통합과정 22학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), 최민규 (한양대학교 의학대학 졸업, +3), Anar Rzayev (KAIST 전산학부 19학번, +3). Late solutions were not graded.

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Solution: 2023-11 Possible outcomes of sums

Let \(S\) be a set of distinct \(20\) integers. A set \(T_A\) is defined as \(T_A:=\{ s_1+s_2+s_3 \mid s_1, s_2, s_3 \in S\}\). What is the smallest possible cardinality of \(T_A\)?

The best solution was submitted by 정희승 (서울대학교 물리천문학부, +4). Congratulations!

Here is the best solution of problem 2023-11.

Other solutions were submitted by 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (새내기과정학부 23학번, +3), 박준성 (KAIST 수리과학과 석박통합과정 22학번, +3), 신민서(KAIST 수리과학과 20학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), 최민규 (한양대학교 의학대학 졸업, +3), Eun Song (+3), James Hamilton Clerk (+3), Anar Rzayev (KAIST 전산학부 19학번, +2), 김준홍 (KAIST 수리과학과 20학번, +2), 최백규 (KAIST 수리과학과 석박통합과정 21학번, +3).

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2023-12 Pairs promoting diversity

Let \(p\) be a prime number at least three and let \(k\) be a positive integer smaller than \(p\). Given \(a_1,\dots, a_k\in \mathbb{F}_p\) and distinct elements \(b_1,\dots, b_k\in \mathbb{F}_p\), prove that there exists a permutation \(\sigma\) of \([k]\) such that the values of \(a_i + b_{\sigma(i)}\) are distinct modulo \(p\).

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Solution: 2023-10 A pair of primes

Find all pairs of prime numbers \( (p, q) \) such that \( pq \) divides \( p^p + q^q + 1 \).

The best solution was submitted by 최백규 (KAIST 생명과학과 박사과정 20학번, +4). Congratulations!

Here is the best solution of problem 2023-10.

Other solutions were submitted by 김준홍 (KAIST 수리과학과 20학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 박준성 (KAIST 수리과학과 석박통합과정 22학번, +3), 신민서(KAIST 수리과학과 20학번, +3), 전해구 (KAIST 기계공학과 졸업, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +3), Matthew Seok (+3).

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