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

Join our FREE webinar on May 1st to learn about managing anxiety.
real analysis
B
No tags match your search
M
G
Topic
First Poster
Last Poster
H
Algebraic Manipulation
Darealzolt   1
N 2 hours ago by Soupboy0
Find the number of pairs of real numbers $a, b, c$ that satisfy the equation $a^4 + b^4 + c^4 + 1 = 4abc$.
1 reply
Darealzolt
4 hours ago
Soupboy0
2 hours ago
J
H
BrUMO 2025 Team Round Problem 13
lpieleanu   1
N 2 hours ago by vanstraelen
Let $\omega$ be a circle, and let a line $\ell$ intersect $\omega$ at two points, $P$ and $Q.$ Circles $\omega_1$ and $\omega_2$ are internally tangent to $\omega$ at points $X$ and $Y,$ respectively, and both are tangent to $\ell$ at a common point $D.$ Similarly, circles $\omega_3$ and $\omega_4$ are externally tangent to $\omega$ at $X$ and $Y,$ respectively, and are tangent to $\ell$ at points $E$ and $F,$ respectively.

Given that the radius of $\omega$ is $13,$ the segment $\overline{PQ}$ has a length of $24,$ and $YD=YE,$ find the length of segment $\overline{YF}.$
1 reply
lpieleanu
Apr 27, 2025
vanstraelen
2 hours ago
J
H
Inequlities
sqing   33
N 3 hours ago by sqing
Let $ a,b,c\geq 0 $ and $ a^2+ab+bc+ca=3 .$ Prove that$$\frac{1}{1+a^2}+ \frac{1}{1+b^2}+ \frac{1}{1+c^2} \geq \frac{3}{2}$$$$\frac{1}{1+a^2}+ \frac{1}{1+b^2}+ \frac{1}{1+c^2}-bc \geq -\frac{3}{2}$$
33 replies
sqing
Jul 19, 2024
sqing
3 hours ago
J
H
Very tasteful inequality
tom-nowy   1
N 3 hours ago by sqing
Let $a,b,c \in (-1,1)$. Prove that $$(a+b+c)^2+3>(ab+bc+ca)^2+3(abc)^2.$$
1 reply
tom-nowy
Today at 10:47 AM
sqing
3 hours ago
J
H
Inequalities
sqing   8
N 3 hours ago by sqing
Let $x\in(-1,1). $ Prove that
$$ \dfrac{1}{\sqrt{1-x^2}} + \dfrac{1}{2+ x^2} \geq \dfrac{3}{2}$$$$ \dfrac{2}{\sqrt{1-x^2}} + \dfrac{1}{1+x^2} \geq 3$$
8 replies
sqing
Apr 26, 2025
sqing
3 hours ago
J
H
đề hsg toán
akquysimpgenyabikho   1
N 5 hours ago by Lankou
làm ơn giúp tôi giải đề hsg

1 reply
akquysimpgenyabikho
Apr 27, 2025
Lankou
5 hours ago
J
H
Inequalities
sqing   2
N Today at 10:05 AM by sqing
Let $a,b,c> 0$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1.$ Prove that
$$ (1-abc) (1-a)(1-b)(1-c) \ge 208 $$$$ (1+abc) (1-a)(1-b)(1-c) \le -224 $$$$(1+a^2b^2c^2) (1-a)(1-b)(1-c) \le -5840 $$
2 replies
sqing
Jul 12, 2024
sqing
Today at 10:05 AM
J
H
9 Physical or online
wimpykid   0
Today at 6:49 AM
Do you think the AoPS print books or the online books are better?

0 replies
wimpykid
Today at 6:49 AM
0 replies
J
H
Three variables inequality
Headhunter   6
N Today at 6:08 AM by lbh_qys
$\forall a\in R$ ,$~\forall b\in R$ ,$~\forall c \in R$
Prove that at least one of $(a-b)^{2}$, $(b-c)^{2}$, $(c-a)^{2}$ is not greater than $\frac{a^{2}+b^{2}+c^{2}}{2}$.

I assume that all are greater than it, but can't go more.
6 replies
Headhunter
Apr 20, 2025
lbh_qys
Today at 6:08 AM
J
H
Sequence
lgx57   8
N Today at 5:08 AM by Vivaandax
$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$. Find the general term of $\{a_n\}$.
8 replies
lgx57
Apr 27, 2025
Vivaandax
Today at 5:08 AM
J
H
J
H
Nice limit
Snoop76   2
N Jan 8, 2025 by Snoop76
Source: Own
If $a_n={\sqrt[n]{n!!}}$ $,$$ $ find :$\lim_{n \to \infty} \sqrt{n}|a_{n+1}+a_{n-1}-2a_n|$.
2 replies
Snoop76
Nov 11, 2024
Snoop76
Jan 8, 2025
J
Nice limit
G H J
Reveal topic tags
double factorial
real analysis
L
G H BBookmark kLocked kLocked NReply
Source: Own
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Snoop76
322 posts
Snoop76
#1 h  Nov 11, 2024, 5:17 PM
Y by
If $a_n={\sqrt[n]{n!!}}$ $,$$ $ find :$\lim_{n \to \infty} \sqrt{n}|a_{n+1}+a_{n-1}-2a_n|$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Snoop76
322 posts
Snoop76
#2 h  Nov 15, 2024, 6:59 AM
Y by
Hint:
$(2n-1)!! = \frac{(2n)!}{(2n)!!} = \sqrt{2}\left(\frac{2n}{e}\right)^{n} \left(1 + O(\frac1{n})\right)$

$(2n)!! = 2^n n! = \sqrt{2n\pi}\left(\frac{2n}{e}\right)^{n} \left(1 +O(\frac1{n})\right)$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Snoop76
322 posts
Snoop76
#3 h  Jan 8, 2025, 8:38 AM
Y by
bumpppp!
Z K Y
N Quick Reply
G
H
=
a