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

We have your learning goals covered with Spring and Summer courses available. Enroll today!
algebra
B
No tags match your search
M
G
Topic
First Poster
Last Poster
H
Inequalities from SXTX
sqing   12
N an hour ago by sqing
T702. Let $ a,b,c>0 $ and $ a+2b+3c=\sqrt{13}. $ Prove that $$ \sqrt{a^2+1} +2\sqrt{b^2+1} +3\sqrt{c^2+1} \geq 7$$S
T703. Let $ a,b $ be real numbers such that $ a+b\neq 0. $. Find the minimum of $ a^2+b^2+(\frac{1-ab}{a+b} )^2.$
T704. Let $ a,b,c>0 $ and $ a+b+c=3. $ Prove that $$ \frac{a^2+7}{(c+a)(a+b)} + \frac{b^2+7}{(a+b)(b+c)} +\frac{c^2+7}{(b+c)(c+a)} \geq 6$$S
12 replies
sqing
Feb 18, 2025
sqing
an hour ago
J
H
Math Competitions
anishka14   5
N Today at 4:33 AM by iwillregretthisnamelater
Hi everyone!

So I am currently in grade 6, and if anyone could give any tips for getting high scores in math competition, that would be great!

I haven't been doing so well in AMC 8, and other competitions like Math Kangaroo, etc....

I feel like i'm stuck, so if anyone could give any resources that helped you learn and score better, could you share that with me?

Thank you so much!

( also how much time should i spend on math every day? )
5 replies
anishka14
Yesterday at 7:55 PM
iwillregretthisnamelater
Today at 4:33 AM
J
H
Something Horrible-a Challenge
Xueshuxue   19
N Today at 3:09 AM by pieMax2713
Hello, I was wondering if it's possible to make 8 with the numbers 5, 3, 5, and 7 under the following rules:
-You can only use 5 twice, 3 once, and 7 once.
-You must use all the numbers.
You can stack numbers to form larger numbers (example: I could take 3 and 5 and turn it into 35 or 53, or use 7, 3, and 5 to make 375.)
-You are allowed to use parentheses.
(Also, I already found out that no 3 digital numbers will work for the solution.)
19 replies
Xueshuxue
Friday at 7:09 PM
pieMax2713
Today at 3:09 AM
J
H
The daily problem!
Leeoz   48
N Today at 3:04 AM by pieMax2713
Every day, I will try to post a new problem for you all to solve! If you want to post a daily problem, you can! :)

Please hide solutions and answers, hints are fine though! :)

The first problem is:
[quote=March 21st Problem]Alice flips a fair coin until she gets 2 heads in a row, or a tail and then a head. What is the probability that she stopped after 2 heads in a row? Express your answer as a common fraction.[/quote]

Past Problems!
48 replies
Leeoz
Mar 21, 2025
pieMax2713
Today at 3:04 AM
J
H
I cannot into nationals, what should I do?
AMathCountsguy10   101
N Today at 2:41 AM by hashbrown2009
I am an eighth grader in North Carolina and I am sort of pressured to make nats this year, I only got 26 on last year's state test (orz to @mathprodigy2011) and I have mocked 39, 40, and 37 on the 2014, 2015, and 2022 tests respectively. I am aiming for a 40 this year because that seems to be the cutoff. How should I sufficiently improve? My test is on March 14th, 2025.

To be more clear, last year I got 16/10 and on the tests I have mocked, I got 27/12, 26/14, and 25/12.
101 replies
AMathCountsguy10
Feb 21, 2025
hashbrown2009
Today at 2:41 AM
J
H
How to pronounce Rusczyk
Dream9   2
N Today at 2:13 AM by KF329
I've always said like ruzz icks. I don't know why I say it as plural but does anyone know how to say the sigma's last name?
2 replies
Dream9
Yesterday at 11:48 PM
KF329
Today at 2:13 AM
J
H
How much does math help?
Dream9   5
N Today at 1:21 AM by yaxuan
Does the math many people on AOPS work for really make a large impact? I get that this is a place for math but I think I'm losing touch with my prior math skill. AMC and AIME are solving problems that have answers and just take practice but math as a job or just like an accountant doesn't need/needs a lot more than this type of math. I might switch to another stem category for college apps pathway. Also, many people who are already very good at competition math don't get good offers because there's people who are even better(note: i get the point of math isn't only college). I think if I work *hard* I'll be AIME qual but I think many people would have higher achievements with the same amount of work in another category. I think competition math really just trains for problem solving, unique thinking, and pattern recognition(aka IQ pts).
5 replies
Dream9
Today at 12:49 AM
yaxuan
Today at 1:21 AM
J
H
problemo
hashbrown2009   1
N Today at 1:03 AM by Dream9
if x/(3^3+4^3) + y/(3^3+6^3) =1

and

x/(5^3+4^3) + y/(5^3+6^3) =1

find the 2 values of x and y.
1 reply
hashbrown2009
Today at 12:49 AM
Dream9
Today at 1:03 AM
J
H
Mathcounts State Sprint Scores
kp34912   31
N Today at 12:42 AM by hashbrown2009
I am practicing previous years' MathCount papers, but they seem to get progressively harder each year.

So, for those of you who scored well in Mathcounts, how many sprint questions are/were you able to do in State/Nationals?

I want to get a sense of where I stand.

Thanks in advance!
31 replies
1 viewing
kp34912
Mar 27, 2025
hashbrown2009
Today at 12:42 AM
J
H
Bogus Proof Marathon
pifinity   7533
N Today at 12:30 AM by Inaaya
Hi!
I'd like to introduce the Bogus Proof Marathon.

In this marathon, simply post a bogus proof that is middle-school level and the next person will find the error. You don't have to post the real solution :P

Use classic Marathon format: 
[hide=P#]a1b2c3[/hide]
[hide=S#]a1b2c3[/hide]


Example posts:

P(x)
-----
S(x)
P(x+1)
-----
Let's go!! Just don't make it too hard!
7533 replies
pifinity
Mar 12, 2018
Inaaya
Today at 12:30 AM
J
H
A retirement to be celebrated...
littlebigbull   23
N Yesterday at 11:37 PM by joyjoyy1
Hello Middle School Math!
In case you have not seen the post in the global announcements, Richard Rusczyk is retiring his position as the founder of AOPS!
I would like to respect him in this thread because without him, there would not be a friendly place to chat about math problems, and make friends!

Thank you!


----littlebigbull
23 replies
littlebigbull
Mar 25, 2025
joyjoyy1
Yesterday at 11:37 PM
J
H
J
H
Equation
algharib   6
N Apr 18, 2016 by Virgil Nicula
HI
Solve this equation :
6 replies
algharib
Mar 15, 2016
Virgil Nicula
Apr 18, 2016
J
Equation
G H J
Reveal topic tags
algebra
L
G H BBookmark kLocked kLocked NReply
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
algharib
18 posts
algharib
#1 h  Mar 15, 2016, 9:04 PM • 2 Y
Y by Adventure10, Mango247
HI
Solve this equation :
Attachments:
This post has been edited 1 time. Last edited by algharib, Mar 15, 2016, 9:08 PM
Reason: equatin
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
algebra_star1234
2467 posts
algebra_star1234
#2 h  Mar 15, 2016, 9:08 PM • 2 Y
Y by Adventure10, Mango247
algharib wrote:
HI
Solve this equation :
$z^{2}+2i\overline{z}=0$
$z\in \mathbb{C}$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
mathguy5041
2659 posts
mathguy5041
#3 h  Mar 15, 2016, 9:15 PM • 2 Y
Y by Adventure10, Mango247
Set $z=a+ib$ then complex quadratic bash? Also note that $0$ works.
$a^2-b^2+2abi+2i(a-ib)=0$, then simplify, and equate real and imaginary parts.
This post has been edited 1 time. Last edited by mathguy5041, Mar 15, 2016, 9:17 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
rlzhang
258 posts
rlzhang
#4 h  Mar 15, 2016, 9:19 PM • 1 Y
Y by Adventure10
Hint
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
FlyingCucumber
1675 posts
FlyingCucumber
#5 h  Mar 15, 2016, 9:47 PM • 2 Y
Y by Adventure10, Mango247
rlzhang wrote:
Hint

No, $z^2$ isn't necessarily $0$ and $2i\overline{z}$ is not necessarily $0$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
mathguy5041
2659 posts
mathguy5041
#6 h  Mar 15, 2016, 10:04 PM • 3 Y
Y by speck, Adventure10, Mango247
mathguy5041 wrote:
Set $z=a+ib$ then complex quadratic bash? Also note that $0$ works.
$a^2-b^2+2abi+2i(a-ib)=0$, then simplify, and equate real and imaginary parts.

continuing
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Virgil Nicula
7054 posts
Virgil Nicula
#7 h  Apr 18, 2016, 1:59 PM • 2 Y
Y by Adventure10, Mango247
See PP5 from here
This post has been edited 1 time. Last edited by Virgil Nicula, Apr 18, 2016, 2:00 PM
Z K Y
N Quick Reply
G
H
=
a