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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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0 replies
jlacosta
Apr 2, 2025
0 replies
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Integration Bee Kaizo
Calcul8er   51
N an hour ago by BaidenMan
Hey integration fans. I decided to collate some of my favourite and most evil integrals I've written into one big integration bee problem set. I've been entering integration bees since 2017 and I've been really getting hands on with the writing side of things over the last couple of years. I hope you'll enjoy!
51 replies
Calcul8er
Mar 2, 2025
BaidenMan
an hour ago
J
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Equation over a finite field
loup blanc   0
3 hours ago
Find the set of $x\in\mathbb{F}_{5^5}$ such that the equation in the unknown $y\in \mathbb{F}_{5^5}$:
$x^3y+y^3+x=0$ admits $3$ roots: $a,a,b$ s.t. $a\not=b$.
0 replies
loup blanc
3 hours ago
0 replies
J
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interesting integral
Martin.s   1
N Today at 2:46 PM by ysharifi
$$\int_0^\infty \frac{\sinh(t)}{t \cosh^3(t)} dt$$
1 reply
Martin.s
Yesterday at 3:12 PM
ysharifi
Today at 2:46 PM
J
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Two times derivable real function
Valentin Vornicu   10
N Today at 2:04 PM by Rohit-2006
Source: RMO 2008, 11th Grade, Problem 3
Let $ f: \mathbb R \to \mathbb R$ be a function, two times derivable on $ \mathbb R$ for which there exist $ c\in\mathbb R$ such that
\[ \frac { f(b)-f(a) }{b-a} \neq f'(c) ,\] for all $ a\neq b \in \mathbb R$.

Prove that $ f''(c)=0$.
10 replies
Valentin Vornicu
Apr 30, 2008
Rohit-2006
Today at 2:04 PM
J
No more topics!
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J
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complex roots of polynomial and inequality for coefficients
su7e   5
N May 17, 2021 by XbenX
polynomial $P(z)=a_nz^n+\dots+ a_1z+a_0\in\mathbb{R}[x]$, $\deg P=n\ge 3$, has all complex roots in a set $\{z\in\mathbb{C}: \mbox{Re}\,z<0\}$.
show that $a_ka_{k+3}<a_{k+1}a_{k+2}$ for $k=0,1,\dots, n-3$.
5 replies
su7e
Nov 29, 2012
XbenX
May 17, 2021
J
complex roots of polynomial and inequality for coefficients
G H J
Reveal topic tags
algebra
polynomial
inequalities
complex analysis
complex analysis unsolved
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G H BBookmark kLocked kLocked NReply
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su7e
16 posts
su7e
#1 h  Nov 29, 2012, 11:42 PM • 2 Y
Y by Adventure10, Mango247
polynomial $P(z)=a_nz^n+\dots+ a_1z+a_0\in\mathbb{R}[x]$, $\deg P=n\ge 3$, has all complex roots in a set $\{z\in\mathbb{C}: \mbox{Re}\,z<0\}$.
show that $a_ka_{k+3}<a_{k+1}a_{k+2}$ for $k=0,1,\dots, n-3$.
Z K Y
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CJA
155 posts
CJA
#2 h  Nov 13, 2019, 1:32 AM • 2 Y
Y by Adventure10, Mango247
any ideas?
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GreenKeeper
1688 posts
GreenKeeper
#3 h  Nov 13, 2019, 6:41 AM • 1 Y
Y by Adventure10
It's probably related to Routh-Hurwitz theorem.

https://en.wikipedia.org/wiki/Hurwitz_matrix
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CJA
155 posts
CJA
#4 h  Nov 13, 2019, 11:52 AM • 1 Y
Y by Adventure10
GreenKeeper wrote:
It's probably related to Routh-Hurwitz theorem.

https://en.wikipedia.org/wiki/Hurwitz_matrix

thank you,this problem is really related to Routh-Hurwitz theorem,it is about Hurwitz polynomial,and I know the problem is called xie xvkai stability criterion（proposed in 1963） now.
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Xiaolin
36 posts
Xiaolin
#6 h  May 17, 2021, 6:44 AM
Y by
Any solutions?
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XbenX
590 posts
XbenX
#7 h  May 17, 2021, 7:20 PM • 1 Y
Y by GreenKeeper
It is IMC 2003 P6; you can find a solution on the official page.
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