Let \(n\) be a fixed positive integer and let \(p\in (0,1)\). Let \(D_n\) be the determinant of a random \(n\times n\) 0-1 matrix whose entries are independent identical random variables, each of which is 1 with the probability \(p\) and 0 with the probability \(1-p\).  Find the expected value and variance of \(D_n\).

The best solution was submitted by Myeongjae Lee (이명재), 2012학번. Congratulations!

Here is his Solution of Problem 2012-21.

Alternative solutions were submitted by 박민재 (2011학번, +3), 김태호 (수리과학과 2011학번, +3), 임현진 (물리학과 2010학번, +3), 김지홍 (수리과학과 2007학번, +2), 서기원 (수리과학과 2009학번, +2).

Let \(A=(a_{ij})\) be an \(n\times n\) upper triangular matrix such that \[a_{ij}=\binom{n-i+1}{j-i}\] for all \(i\le j\). Find the inverse matrix of \(A\).

The best solution was submitted by Minjae Park (박민재), 2011학번. Congratulations!

Here is his Solution of Problem 2012-20.

Alternative solutions were submitted by 서기원 (수리과학과 2009학번, +3), 이명재 (2012학번, +3), 김태호 (수리과학과 2011학번, +3), 임현진 (물리학과 2010학번, +3), 박훈민 (대전과학고 2학년, +3), 윤성철 (홍익대학교 수학교육학과 2009학번, +3), 어수강 (서울대학교 수리과학부 석사과정, +3).

Let \(a_0=3\) and \(a_{n}=a_{n-1}+\sqrt{a_{n-1}^2+3}\) for all \(n\ge 1\). Determine \[\lim_{n\to\infty}\frac{a_n}{2^n}.\]

The best solution was submitted by Myeongjae Lee (이명재), 2012학번. Congratulations!

Here is his Solution of Problem 2012-19.

Alternative solutions were submitted by 박민재 (2011학번, +3), 김태호 (수리과학과 2011학번, +3). Two incorrect solutions were submitted (YSC, KJW).

Let \(r_1,r_2,r_3,\ldots\) be a sequence of all rational numbers in \( (0,1) \) except finitely many numbers. Let \(r_j=0.a_{j,1}a_{j,2}a_{j,3}\cdots\) be a decimal representation of \(r_j\). (For instance, if \(r_1=\frac{1}{3}=0.333333\cdots\), then \(a_{1,k}=3\) for any \(k\).)

Prove that the number \(0.a_{1,1}a_{2,2}a_{3,3}a_{4,4}\cdots\) given by the main diagonal cannot be a rational number.

The best solution was submitted by Kim, Joo Wan (김주완, 수리과학과 2010학번). Congratulations!

Here is his Solution of Problem 2012-18.

Alternative solutions were submitted by 이명재 (2012학번, +3), 김태호 (수리과학과 2011학번, +3), 임현진 (물리학과 2010학번, +3), 박민재 (2011학번, +3), 서기원 (수리과학과 2009학번, +2), 이신영 (2012학번, +2),  윤영수 (2011학번, +2), 박훈민 (대전과학고 2학년, +3), 어수강 (서울대학교 수리과학부 석사과정, +2), 윤성철 (홍익대학교 수학교육학과 2009학번, +2). There were 3 incorrect solutions submitted (JWS, KDR, JSH).

Let \(m\) and \(n\) be odd integers. Determine \[ \sum_{k=1}^\infty \frac{1}{k^2}\tan\frac{k\pi}{m}\tan \frac{k\pi}{n}.\]

The best solution was submitted by Suh, Gee Won (서기원), 수리과학과 2009학번. Congratulations!

Here is his Solution of Problem 2012-17.

One incorrect solution was received (PHM).

Prove that if a finite ring has two elements \(x\) and \(y\) such that \(xy^2=y\), then \( yxy=y\).

The best solution was submitted by Myeongjae Lee (이명재), 2012학번. Congratulations!

Here is Solution of Problem 2012-16.

Alternative solutions were submitted by 김주완 (수리과학과 2010학번, +3), 김지원 (수리과학과 2010학번, +3), 서기원 (수리과학과 2009학번, +3), 김태호 (수리과학과 2011학번, +3), 임현진 (물리학과 2010학번, +3), 박민재 (2011학번, +3), 조상흠 (수리과학과 2010학번, +3), 정우석 (서강대 수학과 2011학번, +3). One incorrect solution (KHK) was submitted.

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).

Determine all continuous functions \(f:(0,\infty)\to(0,\infty)\) such that \[ \int_t^{t^3} f(x) \, dx = 2\int_1^t f(x)\,dx\] for all \(t>0\).

The best solution was submitted by Junghwan Lim (임정환), 수리과학과 2009학번. Congratulations!

Here is his Solution of Problem 2012-14.

Alternative solutions were submitted by 김주완 (2010학번, +3), 김태호 (수리과학과 2011학번, +3), 김홍규 (2011학번, 3), 곽걸담 (물리학과 2011학번, +3), 이신영 (2012학번, +3), 박민재 (2011학번, +3), 박종호 (수리과학과 2009학번, +3), 서기원 (수리과학과 2009학번, +3), 윤영수 (2011학번, +3), 이명재 (2012학번, +3), 조상흠 (2010학번, +3), 조준영 (2012학번, +3), 양지훈 (수리과학과 2010학번, +2), 최원준 (물리학과 2009학번, +2), 장영재 (수리과학과 2011학번, +2), 김건수 (서울대학교 전기컴퓨터공학부 2012학번, +3), 고재윤 (연세대학교, +3), 박훈민 (대전과학고 3학년, +3), 박항 (한국과학영재학교 2010학번, +3), 어수강 (서울대학교 수리과학부 대학원생, +3). There were 3 incorrect solutions submitted (RJH, KDR, JWS).

Let A be a finite set of complex numbers. Prove that there exists a subset B of A such that \[ \bigl\lvert\sum_{z\in B} z\bigr\lvert \ge \frac{ 1}{\pi}\sum_{z\in A} \lvert z\rvert.\]

The best solution was submitted by Minjae Park (박민재), 2011학번. Congratulations!

Here is Solution of Problem 2012-12.

Two incorrect solutions were submitted (M.J.L., W.S.J.).

Let f be a continuous function from [0,1] such that f([0,1]) is a circle. Prove that there exists two closed intervals \(I_1, I_2 \subseteq [0,1]\) such that \(I_1\cap I_2\) has at most one point, \(f(I_1)\) and \(f(I_2)\) are semicircles, and \(f(I_1)\cup f(I_2)\) is a circle.

The best solution was submitted by Myeongjae Lee (이명재), 2012학번. Congratulations!

Here is his Solution of Problem 2012-11.

Alternative solution was submitted by 박민재(2011학번, +3). Two incorrect solutions were submitted (W.S.J., K.M.P.).