For \( a > b > 0 \), find the value of

\[

\int_0^{\infty} \frac{e^{ax} – e^{bx}}{x(e^{ax}+1)(e^{bx}+1)} dx.

\]

**GD Star Rating**

*loading...*

For \( a > b > 0 \), find the value of

\[

\int_0^{\infty} \frac{e^{ax} – e^{bx}}{x(e^{ax}+1)(e^{bx}+1)} dx.

\]

Find the minimum \(m\) (if it exists) such that every convex function \(f:[-1,1]\to[-1,1]\) has a constant \(c\) such that \[ \int_{-1}^1 \lvert f(x)-c\rvert \,dx \le m.\]

The best solution was submitted by Daeseok Lee (이대석, 수리과학과 2017학번). Congratulations!

Here is his solution of problem 2018-16.

Alternative solutions were submitted by 이본우 (수리과학과 2017학번, +3), 이재우 (함양고등학교 3학년, +2).

Find the minimum \(m\) (if it exists) such that every convex function \(f:[-1,1]\to[-1,1]\) has a constant \(c\) such that \[ \int_{-1}^1 \lvert f(x)-c\rvert \,dx \le m.\]

Let \( n \) be a positive integer. Suppose that \( a_1, a_2, \dots, a_n \) are non-zero integers and \( b_1, b_2, \dots, b_n\) are positive integers such that \( (b_i, b_n) = 1 \) for \( i = 1, 2, \dots, n-1 \). Prove that the Diophantine equation

\[

a_1 x_1^{b_1} + a_2 x_2^{b_2} + \dots + a_n x_n^{b_n} = 0

\]

has infinitely many integer solutions \( (x_1, x_2, \dots, x_n) \).

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

Here is his solution of problem 2018-15.

An alternative solution was submitted by Saba Dzmanashvili (수리과학과 2017학번, +3), 강한필 (전산학부 2016학번, +3), 권홍 (중앙대 물리학과, +3), 이본우 (수리과학과 2017학번, +3), 이재우 (함양고등학교 3학년, +3), 최백규 (생명과학과 2016학번, +3), 길현준 (2018학번, +2), 김태균 (수리과학과 2016학번, +2).

Let \( n \) be a positive integer. Suppose that \( a_1, a_2, \dots, a_n \) are non-zero integers and \( b_1, b_2, \dots, b_n\) are positive integers such that \( (b_i, b_n) = 1 \) for \( i = 1, 2, \dots, n-1 \). Prove that the Diophantine equation

\[

a_1 x_1^{b_1} + a_2 x_2^{b_2} + \dots + a_n x_n^{b_n} = 0

\]

has infinitely many integer solutions \( (x_1, x_2, \dots, x_n) \).

Suppose that the edges of a graph \(G\) can be colored by 3 colors so that there is no monochromatic cycle. Prove or disprove that \(G\) has two planar subgraphs \(G_1,G_2\) such that \(E(G)=E(G_1)\cup E(G_2)\).

The best solution was submitted by Huy Tùng Nguyễn (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2018-14.

Alternative solutions were submitted by 강한필 (전산학부 2016학번, +3, solution) and 김일희 (수리과학과 2001학번 동문, +3, solution). There was one incorrect submission.

At the moment, this problem remains open.

Suppose that the edges of a graph \(G\) can be colored by 3 colors so that there is no monochromatic cycle. Prove or disprove that \(G\) has two planar subgraphs \(G_1,G_2\) such that \(E(G)=E(G_1)\cup E(G_2)\).

Hint: The answer is NO. Disprove it.

Suppose that the edges of a graph \(G\) can be colored by 3 colors so that there is no monochromatic cycle. Prove or disprove that \(G\) has two planar subgraphs \(G_1,G_2\) such that \(E(G)=E(G_1)\cup E(G_2)\).

Assume that \( x \in \mathbb{R}^n \) with at least \( k \) non-zero entries \( ( k> 0 ) \). Let

\[

A = \{ y \in \{-1, 1\}^n : y \cdot x = 0 \}.

\]

Prove that \( |A| \leq k^{-1/2} 2^n \).

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

Here is his solution of problem 2018-13.

An alternative solution was submitted by 이대석 (수리과학과 2017학번, +3). Two incorrect solutions were received.