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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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[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
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thanks u!
Ruji2018252   1
N 7 minutes ago by arqady
Let $a,b,c>2$ and $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=a+b+c-8$. Prove:
\[ab+bc+ac\leqslant 27\]
1 reply
1 viewing
Ruji2018252
Yesterday at 6:00 PM
arqady
7 minutes ago
J
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Values of p(1)
uNc   11
N 17 minutes ago by Nari_Tom
Source: Baltic way 2009
A polynomial $p(x)$ of degree $n\ge 2$ has exactly $n$ real roots, counted with multiplicity. We know that the coefficient of $x^n$ is $1$, all the roots are less than or equal to $1$, and $p(2)=3^n$. What values can $p(1)$ take?
11 replies
uNc
Nov 11, 2009
Nari_Tom
17 minutes ago
J
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Perpendicularity
April   30
N 17 minutes ago by Tsikaloudakis
Source: CGMO 2007 P5
Point $D$ lies inside triangle $ABC$ such that $\angle DAC = \angle DCA = 30^{\circ}$ and $\angle DBA = 60^{\circ}$. Point $E$ is the midpoint of segment $BC$. Point $F$ lies on segment $AC$ with $AF = 2FC$. Prove that $DE \perp EF$.
30 replies
April
Dec 28, 2008
Tsikaloudakis
17 minutes ago
J
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Prove that x1=x2=....=x2025
Rohit-2006   1
N 19 minutes ago by flower417477
Source: A mock
The real numbers $x_1,x_2,\cdots,x_{2025}$ satisfy,
$$x_1+x_2=2\bar{x_1}, x_2+x_3=2\bar{x_2},\cdots, x_{2025}+x_1=2\bar{x_{2025}}$$Where {$\bar{x_1},\cdots,\bar{x_{2025}}$} is a permutation of $x_1,x_2,\cdots,x_{2025}$. Prove that $x_1=x_2=\cdots=x_{2025}$
1 reply
Rohit-2006
3 hours ago
flower417477
19 minutes ago
J
No more topics!
H
J
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Inspired by old results
sqing   2
N Apr 3, 2025 by sqing
Source: Own
Let $ a,b,c \ge \frac{1}{21} $ and $ a+b+c=1. $. Prove that
$$(a^2-ab+b^2)(b^2-bc+c^2)(c^2-ca+a^2)\geq\frac{1}{729} $$Let $ a,b,c \ge \frac{1}{10} $ and $ a+b+c=2. $. Prove that
$$(a^2-ab+b^2)(b^2-bc+c^2)(c^2-ca+a^2)\geq\frac{64}{729} $$Let $ a,b,c \ge \frac{1}{11} $ and $ a+b+c=2. $. Prove that
$$(a^2-ab+b^2)(b^2-bc+c^2)(c^2-ca+a^2)\geq \frac{145161}{1771561} $$
2 replies
sqing
Apr 3, 2025
sqing
Apr 3, 2025
J
Inspired by old results
G H J
Reveal topic tags
inequalities proposed
algebra
inequalities
L
G H BBookmark kLocked kLocked NReply
Source: Own
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sqing
41479 posts
sqing
#1 h  Apr 3, 2025, 1:42 PM • 1 Y
Y by cubres
Let $ a,b,c \ge \frac{1}{21} $ and $ a+b+c=1. $. Prove that
$$(a^2-ab+b^2)(b^2-bc+c^2)(c^2-ca+a^2)\geq\frac{1}{729} $$Let $ a,b,c \ge \frac{1}{10} $ and $ a+b+c=2. $. Prove that
$$(a^2-ab+b^2)(b^2-bc+c^2)(c^2-ca+a^2)\geq\frac{64}{729} $$Let $ a,b,c \ge \frac{1}{11} $ and $ a+b+c=2. $. Prove that
$$(a^2-ab+b^2)(b^2-bc+c^2)(c^2-ca+a^2)\geq \frac{145161}{1771561} $$
This post has been edited 1 time. Last edited by sqing, Apr 3, 2025, 1:52 PM
Z K Y
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sqing
41479 posts
sqing
#2 h  Apr 3, 2025, 2:16 PM
Y by
Let $ a,b,c \ge \frac{1}{4} $ and $ a+b+c=3. $. Prove that
$$(a^2- \frac{1}{2} ab+b^2)(b^2-\frac{1}{2} bc+c^2)(c^2-\frac{1}{2} ca+a^2)\geq\frac{27}{8} $$Let $ a,b,c \ge \frac{1}{16} $ and $ a+b+c=3. $. Prove that
$$(a^2- \frac{3}{2} ab+b^2)(b^2-\frac{3}{2} bc+c^2)(c^2-\frac{3}{2} ca+a^2)\geq\frac{1}{8} $$
Z K Y
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sqing
41479 posts
sqing
#3 h  Apr 3, 2025, 2:41 PM
Y by
Let $ a,b,c \ge \frac{1}{6} $ and $ a+b+c=2. $. Prove that
$$(a^2- \frac{1}{2} ab+b^2)(b^2-\frac{1}{2} bc+c^2)(c^2-\frac{1}{2} ca+a^2)\geq\frac{8}{27} $$Let $ a,b,c \ge \frac{1}{24} $ and $ a+b+c=2. $. Prove that
$$(a^2- \frac{3}{2} ab+b^2)(b^2-\frac{3}{2} bc+c^2)(c^2-\frac{3}{2} ca+a^2)\geq\frac{8}{729} $$
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