2019-03 Simple spectrum

Suppose that \( T \) is an \( N \times N \) matrix
\[
T = \begin{pmatrix}
a_1 & b_1 & 0 & \cdots & 0 \\
b_1 & a_2 & b_2 & \ddots & \vdots \\
0 & b_2 & a_3 & \ddots & 0 \\
\vdots & \ddots & \ddots & \ddots & b_{N-1} \\
0 & \cdots & 0 & b_{N-1} & a_N
\end{pmatrix}
\]
with \( b_i > 0 \) for \( i =1, 2, \dots, N-1 \). Prove that \( T \) has \( N \) distinct eigenvalues.

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Solution: 2019-02 Simplification of an expression with factorials

For any positive integers m and n, show that

\[ C_{n,m} = \frac{(mn)!}{(m!)^n n!} \] is an integer.

The best solution was submitted by 이영민 (수리과학과 대학원생). Congratulations!

Here is his solution of problem 2019-02.

Other solutions were submitted by Alfonso Alvarenga (전산학부 2015학번, +3), Sahngyoon Rhee (UCLA, +3), 고성훈 (2018학번, +3), 길현준 (2018학번, +3), 김기수 (수리과학과 2018학번), 김민서 (2019학번, +3), 김태균 (수리과학과 2016학번, +3), 박건규 (수리과학과 2015학번, +3), 박수찬 (전산학부 2017학번, +3), 박현영 (전기및전자공학부 2016학번, +3), 윤현민 (수리과학과 2018학번), 이본우 (수리과학과 2017학번, +3), 이정환 (수리과학과 2015학번, +3), 이종서 (2019학번, +3), 이태영 (수리과학과 졸업생, +3), 조재형 (수리과학과 2016학번, +3), 조정휘 (건국대학교 수학과 2014학번, +3), 채지석 (수리과학과 2016학번, +3), 최백규 (생명과학과 2016학번, +3). Late solutions are not graded.

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Solution: 2019-01 Equilateral polygon

Suppose that \( \Pi \) is a closed polygon in the plane. If \( \Pi \) is equilateral \( k \)-gon, and if \( A \) is the area of \( \Pi \), and \( L \) the length of its boundary, prove that
\[
\frac{A}{L^2} \leq \frac{1}{4k} \cot \frac{\pi}{k} \leq \frac{1}{4\pi}.
\]

The best solution was submitted by 윤창기 (서울대학교 화학과). Congratulations!

Here is his solution of problem 2019-01.

Similar solutions were submitted by 길현준 (2018학번, +3), 조재형 (수리과학과 2016학번, +3), 김태균 (수리과학과 2016학번, +2). Alternative solution was submitted by 고성훈 (2018학번, +3). Four incorrect solutions were received.

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Solution: 2018-23 Game of polynomials

Two players play a game with a polynomial with undetermined coefficients
\[
1 + c_1 x + c_2 x^2 + \dots + c_7 x^7 + x^8.
\]
Players, in turn, assign a real number to an undetermined coefficient until all coefficients are determined. The first player wins if the polynomial has no real zeros, and the second player wins if the polynomial has at least one real zero. Find who has the winning strategy.

The best solution was submitted by Ha, Seokmin (하석민, 수리과학과 2017학번). Congratulations!

Here is his solution of problem 2018-23.

Alternative solutions were submitted by 채지석 (수리과학과 2016학번, +3), 권홍 (중앙대 물리학과, +2).

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2018-23 Game of polynomials

Two players play a game with a polynomial with undetermined coefficients
\[
1 + c_1 x + c_2 x^2 + \dots + c_7 x^7 + x^8.
\]
Players, in turn, assign a real number to an undetermined coefficient until all coefficients are determined. The first player wins if the polynomial has no real zeros, and the second player wins if the polynomial has at least one real zero. Find who has the winning strategy.

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Solution: 2018-22 Two monic quadratic polynomials

Let \(f_1(x)=x^2+a_1x+b_1\) and \(f_2(x)=x^2+a_2x+b_2\) be polynomials with real coefficients. Prove or disprove that the following are equivalent.

(i) There exist two positive reals \(c_1, c_2\) such that \[ c_1f_1(x)+ c_2 f_2(x) > 0\] for all reals \(x\).

(ii) There  is no real \(x\) such that \( f_1(x)\le 0\) and \( f_2(x)\le 0\).

The best solution was submitted by Gil, Hyunjun (길현준, 2018학번). Congratulations!

Here is his solution of problem 2018-22.

Alternative solutions were submitted by 김태균 (수리과학과 2016학번, +3), 서준영 (수리과학과 대학원생, +3), 이본우 (수리과학과 2017학번, +3), 채지석 (수리과학과 2016학번, +3), 하석민 (수리과학과 2017학번, +3), 최백규 (생명과학과 2016학번, +2). There was one incorrect submission.

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2018-22 Two monic quadratic polynomials

Let \(f_1(x)=x^2+a_1x+b_1\) and \(f_2(x)=x^2+a_2x+b_2\) be polynomials with real coefficients. Prove or disprove that the following are equivalent.

(i) There exist two positive reals \(c_1, c_2\) such that \[ c_1f_1(x)+ c_2 f_2(x) > 0\] for all reals \(x\).

(ii) There  is no real \(x\) such that \( f_1(x)\le 0\) and \( f_2(x)\le 0\).

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Solution: 2018-21 AM-GM inequality

Does there exist a (possibly \(n\)-dependent) constant \( C \) such that
\[
\frac{C}{a_n} \sum_{1 \leq i < j \leq n} (a_i-a_j)^2 \leq \frac{a_1+ \dots + a_n}{n} – \sqrt[n]{a_1 \dots a_n} \leq \frac{C}{a_1} \sum_{1 \leq i < j \leq n} (a_i-a_j)^2
\]
for any \( 0 < a_1 \leq a_2 \leq \dots \leq a_n \)?

The best solution was submitted by Jiseok Chae (채지석, 수리과학과 2016학번). Congratulations!

Here is his solution of problem 2018-21.

Alternative solutions were submitted by 하석민 (수리과학과 2017학번, +3),
이본우 (수리과학과 2017학번, +2). One incorrect submission was received.

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