# 2009-11 Circles and lines

Does there exist a set of circles on the plane such that every line intersects at least one but at most 100 of them?

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# Solution: 2009-10 x and cos x

Find all real-valued continuous function f on the reals such that f(x)=f(cos x) for every real number x.

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

Here is his Solution of Problem 2009-10.

Alternative solutions were submitted by 백형렬(수리과학과 2003학번, +3), 김치헌(수리과학과 2006학번, +3), 이재송(전산학과 2006학번, +3), 권상훈(수리과학과 2006학번, +3), 조용화(수리과학과 2006학번, +2), 김일희 & 오성진 (Princeton Univ., Graduate students).

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# 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 One incorrect solution was submitted (0 point) and one (incorrect) solution was submitted but later withdrawn.

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# 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.
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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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