Category Archives: solution

Solution: 2009-9 min or max

Suppose that * is an associative and commutative binary operation on the set of rational numbers such that 

  1. 0*0=0
  2. (a+c)*(b+c)=(a*b)+c for all rational numbers a,b,c.

Prove that either

  1. a*b=max(a,b) for all rational numbers a,b, or
  2. a*b=min(a,b) for all rational number a,b.

The best solution was submitted by Jaesong Lee (이재송), 전산학과 2005학번. Congratulations!

Check his Solution of Problem 2009-9.

Alternative solutions were submitted by 김치헌 (수리과학과 2006학번, +3), 권상훈 (수리과학과 2006학번, +3), 양해훈 (수리과학과 2008학번, +3), 백형렬 (수리과학과 2003학번, +2), 김일희 & 오성진 (Princeton Univ., Graduate Student). One incorrect solution was submitted (0 point) and one (incorrect) solution was submitted but later withdrawn.

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Solution: 2009-8 Fibonacci number divisible by k

Prove that for every positive integer k, there exists a positive Fibonacci number divisible by k.

The best solution was submitted by Hojin Kim (김호진), 2009학번. Congratulations!

Here is his Solution of Problem 2009-8.

There were 6 other solutions submitted by KAIST undergraduates; 조강진 (2009학번), 이재송 (전산학과 2005학번), 백형렬 (수리과학과 2003학번), 조용화 (수리과학과 2006학번), 김치헌 (수리과학과 2006학번), 권상훈 (수리과학과 2006학번). All will receive 3 points each. In addition, there were 3 other correct solutions submitted; 김성윤 (Mathematics, MIT, Undergraduate Class of ’09), 김일희 (PACM, Princeton Univ., Graduate Student), 정준혁 (Mathematics, Princeton Univ., Graduate Student).

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Solution: 2009-7 A rational problem

Let n>1 be an integer and let x>1 be a real number. Prove that if
\(\sqrt[n]{x+\sqrt{x^2-1}}+\sqrt[n]{x-\sqrt{x^2-1}}\)
is a rational number, then x is rational.

The best solution was submitted by Sungyoon Kim (김성윤) (Mathematics, MIT, Class of ’09). Congratulations! (Though, he is not eligible for earning points and taking prizes.)

Here is his Solution of Problem 2009-7.

There were 5 other solutions submitted: 김호진 (2009학번), 백형렬 (수리과학과 2003학번), 이재송 (전산학과 2005학번), 조강진 (2009학번), 박승균 (수리과학과 2008학번). All will receive 3 points each.

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Solution: 2009-6 Sum of integers of the fourth power

 

Prove that each positive integer x can be written as a sum of at most 53 integers to the fourth power. In other words, for every positive integer x, there exist \(y_1,y_2,\ldots,y_k\) with \(k\le 53\) such that \(x=\sum_{i=1}^k y_i^4\).

The best solution was submitted by SangHoon Kwon (권상훈), 수리과학과 2006학번.

Here is his Solution of Problem 2009-6.

There were 4 other submitted solutions: 백형렬, 이재송, 조강진, 김치헌 (+3).

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Solution: 2009-5 Random points and the origin

 

If we choose n points on a circle randomly and independently with uniform distribution, what is the probability that the center of the circle is contained in the interior of the convex hull of these n points?

The best solution was submitted by Chiheon Kim (김치헌), 수리과학과 2006학번. Congratulations!

There were 2 incorrect solutions submitted.

Click here for his Solution of Problem 2009-5.

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Solution: 2009-4 Initial values

Let \(a_0=a\) and \(a_{n+1}=a_n (a_n^2-3)\). Find all real values \(a\) such that the sequence \(\{a_n\}\) converges.

The best solution was submitted by Jaesong Lee (이재송), 전산학과 2005학번. Congratulations!

Check his Solution of Problem 2009-4. (This proof can be slightly improved in the second half.)

Alternative solutions were submitted by 김치헌 (+2), 백형렬 (+3).

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Solution: 2009-3 Intersecting family

 Let \(\mathcal F\) be a collection of subsets (of size r) of a finite set E such that \(X\cap Y\neq\emptyset\) for all \(X, Y\in \mathcal F\). Prove that there exists a subset S of E such that \(|S|\le (2r-1)\binom{2r-3}{r-1}\) and \(X\cap Y\cap S\neq\emptyset\) for all \(X,Y\in\mathcal F\).

The best solution was submitted by Hyung Ryul Baik (백형렬), 수리과학과 2003학번. Congratulations!

Click here for his Solution of Problem 2009-3.

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Solution: 2009-2 Sequence of Log

Let \(a_1<\cdots\) be a sequence of positive integers such that \(\log a_1, \log a_2,\log a_3,\cdots\) are linearly independent over the rational field \(\mathbb Q\). Prove that \(\lim_{k\to \infty} a_k/k=\infty\).

The best solution was submitted by SangHoon Kwon (권상훈), 수리과학과 2006학번. Congratulations!

Click here for his Solution of Problem 2009-2.

There were 3 other submitted solutions which will earn points: 김치헌+3, 김린기+3,  조강진+2.

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Status: 2009-1 Integer or not

So far 5 solutions were submitted but I am not sure whether any of them is absolutely correct. They (이병찬, 류연식, 권상훈, 김린기, 조강진) will all receive 2 points each.
Here is the origin of typical mistakes: if x|z and y|z, then xy|z.
The problem remains open.

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