# 2018-17 Mathematica does not know the answer

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

# Solution: 2018-16 A convex function

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).

GD Star Rating

# 2018-16 A convex function

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

GD Star Rating

# Solution: 2018-15 Diophantine equation

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).

GD Star Rating

# 2018-15 Diophantine equation

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 $$(x_1, x_2, \dots, x_n)$$.

GD Star Rating

# Solution: 2018-14 Forests and Planes

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.

GD Star Rating

# Status: 2018-14 Forests and Planes

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.

GD Star Rating

# 2018-14 Forests and Planes

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 $$E(G)=E(G_1)\cup E(G_2)$$.

GD Star Rating

# Solution: 2018-13 Bernoulli vectors

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.

GD Star Rating