Consider all non-empty subsets \(S_1,S_2,\ldots,S_{2^n-1}\) of \(\{1,2,3,\ldots,n\}\). Let \(A=(a_{ij})\) be a \((2^n-1)\times(2^n-1)\) matrix such that \[a_{ij}=\begin{cases}1 & \text{if }S_i\cap S_j\ne \emptyset,\\0&\text{otherwise.}\end{cases}\] What is \(\lvert\det A\rvert\)?

The best solution was submitted by Kim, Taeho (김태호), 수리과학과 2011학번. Congratulations!

Here is his Solution of Problem 2012-24.

Alternative solutions were submitted by 이명재 (2012학번, +3), 임현진 (물리학과 2010학번, +3), 정종헌 (2012학번, +2),  어수강 (서울대학교 수리과학부 석사과정, +3).

Let \(n\) be a fixed positive integer. Find all functions \( f:\mathbb{R}\to\mathbb{R}\) satisfying \[ f(x^{n+1}-y^{n+1})=(x-y)[f(x)^n+f(x)^{n-1}f(y)+\cdots+f(x)f(y)^{n-1}+f(y)^n].\]

The best solution was submitted by Kim, Taeho (김태호), 수리과학과 2011학번. Congratulations!

Here is his Solution of Problem 2012-15.

Alternative solutions were submitted by 임정환 (수리과학과 2009학번, +3), 곽걸담 (물리학과 2011학번, +2), 서기원 (수리과학과 2009학번, +2),  김홍규 (수리과학과 2011학번, +2), 김지원 (수리과학과 2010학번, +2), 이명재 (2012학번, +2), 조상흠 (수리과학과 2010학번, +2). There were 2 incorrect submissions (LHJ, KDR).

Let X be the set of all postive real numbers c such that  \[\frac{\prod_{k=1}^{n-1} \sin\left( \frac{k \pi}{2n}\right)}{c^n} \]  converges as n goes to infinity. Find the infimum of X.

The best solution was submitted by Taeho Kim (김태호, 수리과학과 2011학번). Congratulations!

Here is his Solution of Problem 2012-7.

Alternative solutions were submitted by 서기원 (수리과학과 2009학번, +3), 박민재 (2011학번, +3), 조준영 (2012학번, +3), 이명재 (2012학번, +3), 정우석 (서강대 2011학번, +3), 천용 (전남대 의예과 2011학번 +3), 어수강 (서울대학교 석사과정, +2).

Let f(n) be the maximum positive integer m such that the sum of all positive divisors of m is less than or equal to n. Find all positive integers k such that there are infinitely many positive integers n satisfying the equation n-f(n)=k.

The best solution was submitted by Taeho Kim (김태호), 2011학번. Congratulations!

Here is his Solution of Problem 2011-17.

Alternative solutions were submitted by 김범수 (수리과학과 2010학번, +3), 서기원 (수리과학과 2009학번, +3), 박승균 (수리과학과 2008학번, +3), 장경석 (2011학번, +3), 구도완 (해운대고등학교 3학년, +3), 손동현 (유성고등학교 2학년, +2), 어수강 (서울대학교 석사과정, +2).

Update: I forgot to add 최민수 (2011학번, +3) into the list of people submitted alternative solutions.