Art of Problem Solving
AoPS Online AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy AoPS Academy
Small live classes for advanced math
and language arts learners in grades 1-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Stay ahead of learning milestones! Enroll in a class over the summer!
algebra
B
No tags match your search
M
G
Topic
First Poster
Last Poster
H
A+b+c=0
Xixas   17
N 42 minutes ago by AlexCenteno2007
Source: Lithuanian Mathematical Olympiad 2006
Show that if $a+b+c=0$ then $(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b})(\frac{b-c}{a}+\frac{c-a}{b}+\frac{a-b}{c})=9$.
17 replies
Xixas
Apr 12, 2006
AlexCenteno2007
42 minutes ago
J
No more topics!
H
J
H
3 var inquality
sqing   1
N Apr 8, 2025 by hashtagmath
Source: Own
Let $ a,b,c>0 $ and $ \dfrac{a}{bc}+\dfrac{2b}{ca}+\dfrac{5c}{ab}\leq 12.$ Prove that$$ a^2+b^2+c^2\geq 1$$
1 reply
sqing
Apr 6, 2025
hashtagmath
Apr 8, 2025
J
3 var inquality
G H J
Reveal topic tags
inequalities proposed
inequalities
algebra
L
G H BBookmark kLocked kLocked NReply
Source: Own
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
sqing
41636 posts
sqing
#1 h  Apr 6, 2025, 1:11 PM
Y by
Let $ a,b,c>0 $ and $ \dfrac{a}{bc}+\dfrac{2b}{ca}+\dfrac{5c}{ab}\leq 12.$ Prove that$$ a^2+b^2+c^2\geq 1$$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
hashtagmath
1601 posts
hashtagmath
#2 h  Apr 8, 2025, 7:41 PM
Y by
Dear sqing,

I have noticed thousands upon thousands of inequalities that you have posted to HSO and was wondering where you get the inspiration, imagination, and even the validation that such inequalities are true? Also, what do you find particularly appealing and important about specifically inequalities rather than other branches of mathematics?

Thank you :)
Z K Y
N Quick Reply
G
H
=
a