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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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0 replies
jlacosta
May 1, 2025
0 replies
k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
Classic Diophantine
Adywastaken   4
N 27 minutes ago by mrtheory
Source: NMTC 2024/6
Find all natural number solutions to $3^x-5^y=z^2$.
4 replies
Adywastaken
Today at 3:39 PM
mrtheory
27 minutes ago
Where are the Circles?
luminescent   43
N an hour ago by Amkan2022
Source: EGMO 2022/1
Let $ABC$ be an acute-angled triangle in which $BC<AB$ and $BC<CA$. Let point $P$ lie on segment $AB$ and point $Q$ lie on segment $AC$ such that $P \neq B$, $Q \neq C$ and $BQ = BC = CP$. Let $T$ be the circumcenter of triangle $APQ$, $H$ the orthocenter of triangle $ABC$, and $S$ the point of intersection of the lines $BQ$ and $CP$. Prove that $T$, $H$, and $S$ are collinear.
43 replies
luminescent
Apr 9, 2022
Amkan2022
an hour ago
Divisibilty...
Sadigly   0
2 hours ago
Source: Azerbaijan Junior NMO 2025 P2
Find all $4$ consecutive even numbers, such that the square of their product divides the sum of their squares.
0 replies
Sadigly
2 hours ago
0 replies
Quadratic system
juckter   35
N 3 hours ago by shendrew7
Source: Mexico National Olympiad 2011 Problem 3
Let $n$ be a positive integer. Find all real solutions $(a_1, a_2, \dots, a_n)$ to the system:

\[a_1^2 + a_1 - 1 = a_2\]\[ a_2^2 + a_2 - 1 = a_3\]\[\hspace*{3.3em} \vdots \]\[a_{n}^2 + a_n - 1 = a_1\]
35 replies
juckter
Jun 22, 2014
shendrew7
3 hours ago
IMO Shortlist 2012, Geometry 3
lyukhson   75
N 3 hours ago by numbertheory97
Source: IMO Shortlist 2012, Geometry 3
In an acute triangle $ABC$ the points $D,E$ and $F$ are the feet of the altitudes through $A,B$ and $C$ respectively. The incenters of the triangles $AEF$ and $BDF$ are $I_1$ and $I_2$ respectively; the circumcenters of the triangles $ACI_1$ and $BCI_2$ are $O_1$ and $O_2$ respectively. Prove that $I_1I_2$ and $O_1O_2$ are parallel.
75 replies
lyukhson
Jul 29, 2013
numbertheory97
3 hours ago
Diophantine
TheUltimate123   31
N 4 hours ago by SomeonecoolLovesMaths
Source: CJMO 2023/1 (https://aops.com/community/c594864h3031323p27271877)
Find all triples of positive integers \((a,b,p)\) with \(p\) prime and \[a^p+b^p=p!.\]
Proposed by IndoMathXdZ
31 replies
TheUltimate123
Mar 29, 2023
SomeonecoolLovesMaths
4 hours ago
Cyclic ine
m4thbl3nd3r   1
N 4 hours ago by arqady
Let $a,b,c>0$ such that $a^2+b^2+c^2=3$. Prove that $$\sum \frac{a^2}{b}+abc \ge 4$$
1 reply
m4thbl3nd3r
Today at 3:34 PM
arqady
4 hours ago
Non-homogenous Inequality
Adywastaken   7
N 4 hours ago by ehuseyinyigit
Source: NMTC 2024/7
$a, b, c\in \mathbb{R_{+}}$ such that $ab+bc+ca=3abc$. Show that $a^2b+b^2c+c^2a \ge 2(a+b+c)-3$. When will equality hold?
7 replies
Adywastaken
Today at 3:42 PM
ehuseyinyigit
4 hours ago
FE with devisibility
fadhool   2
N 4 hours ago by ATM_
if when i solve an fe that is defined in the set of positive integer i found m|f(m) can i set f(m) =km such that k is not constant and of course it depends on m but after some work i find k=c st c is constant is this correct
2 replies
fadhool
Today at 4:25 PM
ATM_
4 hours ago
Japan MO Finals 2023
parkjungmin   2
N 4 hours ago by parkjungmin
It's hard. Help me
2 replies
parkjungmin
Yesterday at 2:35 PM
parkjungmin
4 hours ago
Iranian geometry configuration
Assassino9931   2
N 4 hours ago by Captainscrubz
Source: Al-Khwarizmi Junior International Olympiad 2025 P7
Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$, such that $CD$ is not a diameter of its circumcircle. The lines $AD$ and $BC$ intersect at point $P$, so that $A$ lies between $D$ and $P$, and $B$ lies between $C$ and $P$. Suppose triangle $PCD$ is acute and let $H$ be its orthocenter. The points $E$ and $F$ on the lines $BC$ and $AD$, respectively, are such that $BD \parallel HE$ and $AC\parallel HF$. The line through $E$, perpendicular to $BC$, intersects $AD$ at $L$, and the line through $F$, perpendicular to $AD$, intersects $BC$ at $K$. Prove that the points $K$, $L$, $O$ are collinear.

Amir Parsa Hosseini Nayeri, Iran
2 replies
Assassino9931
Today at 9:39 AM
Captainscrubz
4 hours ago
f(m + n) >= f(m) + f(f(n)) - 1
orl   30
N 5 hours ago by ezpotd
Source: IMO Shortlist 2007, A2, AIMO 2008, TST 2, P1, Ukrainian TST 2008 Problem 8
Consider those functions $ f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition
\[ f(m + n) \geq f(m) + f(f(n)) - 1
\]
for all $ m,n \in \mathbb{N}.$ Find all possible values of $ f(2007).$

Author: Nikolai Nikolov, Bulgaria
30 replies
orl
Jul 13, 2008
ezpotd
5 hours ago
Add d or Divide by a
MarkBcc168   25
N 6 hours ago by Entei
Source: ISL 2022 N3
Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define
$$x_{k+1} = \begin{cases}
x_k + d &\text{if } a \text{ does not divide } x_k \\
x_k/a & \text{if } a \text{ divides } x_k
\end{cases}$$Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.
25 replies
MarkBcc168
Jul 9, 2023
Entei
6 hours ago
Alice and Bob play, 8x8 table, white red black, minimum n for victory
parmenides51   14
N 6 hours ago by Ilikeminecraft
Source: JBMO Shortlist 2018 C3
The cells of a $8 \times 8$ table are initially white. Alice and Bob play a game. First Alice paints $n$ of the fields in red. Then Bob chooses $4$ rows and $4$ columns from the table and paints all fields in them in black. Alice wins if there is at least one red field left. Find the least value of $n$ such that Alice can win the game no matter how Bob plays.
14 replies
parmenides51
Jul 22, 2019
Ilikeminecraft
6 hours ago
Advanced topics in Inequalities
va2010   23
N Apr 22, 2025 by Novmath
So a while ago, I compiled some tricks on inequalities. You are welcome to post solutions below!
23 replies
va2010
Mar 7, 2015
Novmath
Apr 22, 2025
Advanced topics in Inequalities
G H J
G H BBookmark kLocked kLocked NReply
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va2010
1276 posts
#1 • 11 Y
Y by DrMath, jh235, TheCrafter, Eugenis, mathleticguyyy, somebodyyouusedtoknow, Adventure10, Mango247, NicoN9, cubres, ehuseyinyigit
So a while ago, I compiled some tricks on inequalities. You are welcome to post solutions below!
Attachments:
advanced-topics-inequalities (8).pdf (139kb)
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tastymath75025
3223 posts
#2 • 1 Y
Y by Adventure10
Will edit later maybe

4.1: sol

5.1: sol

[/hide]
This post has been edited 2 times. Last edited by tastymath75025, Mar 7, 2015, 5:31 AM
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Nuran2010
85 posts
#3 • 1 Y
Y by TunarHasanzade
3.1:Click to reveal hidden text
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Nuran2010
85 posts
#4 • 2 Y
Y by TunarHasanzade, Strangett
Similar logic,but a bit different approach for 5.1:
Click to reveal hidden text
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sqing
42105 posts
#5 • 1 Y
Y by Strangett
Problem 5.1. (Tuan Le)
Let $a,b,c$ be positive real numbers such that $  abc\geq 1$ Prove the inequality$$\frac{a}{a^3+b^2+c}+\frac{b}{b^3+c^2+a}+\frac{c}{c^3+a^2+b}\le 1.$$
Problem4.2. (T.Q.Anh)
Let $a,b,c>0$ and $ab+bc+ca=3$.Prove that$$ \frac{a}{2a+b^2}+\frac{b}{2b+c^2}+\frac{c}{2c+a^2} \leq 1$$
Problem 3.1. (Titu Andresscu)
Let $ a,b,c>0 $ and $ a+b+c\geq 3 .$ Prove that$$ \frac{1}{a^2+b+c}+\frac{1}{b^2+c+a}+\frac{1}{c^2+a+b}\leq 1$$https://artofproblemsolving.com/community/c6h1838005p12333824
This post has been edited 2 times. Last edited by sqing, Apr 16, 2025, 2:33 PM
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giangtruong13
145 posts
#6
Y by
You see NOTHING
This post has been edited 1 time. Last edited by giangtruong13, Apr 16, 2025, 1:55 PM
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Primeniyazidayi
98 posts
#7
Y by
The direction of 3.2 is false.Anyway,here is an approach from "Problems from the book,Titu Andreescu and Gabriel Dospinescu"
S3.2
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sqing
42105 posts
#8
Y by
If $ a,b,c$ are reals show that:
$$ \frac{(b+c-a)^2}{(b+c)^2+a^2}+\frac{(c+a-b)^2}{(c+a)^2+b^2}+\frac{(a+b-c)^2}{(a+b)^2+c^2}\geq\frac{3}{5}$$
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sqing
42105 posts
#9
Y by
Let $x,y,z>0$ and $xyz=x+y+z$.Prove that$$\frac{(x-1)^2}{x^2+1}+\frac{(y-1)^2}{y^2+1}+\frac{(z-1)^2}{z^2+1} \geq 3-\frac{3\sqrt 3}{2}$$*
Let $ a, b, c\geq -\frac{3}{4},a+b+c=1 .$ Prove that
$$\frac{(a-1)^{2}}{a^2+1 }+\frac{(b-1)^{2}}{b^2+1 }+\frac{(c-1)^{2}}{c^2+1 }\geq \frac{6}{5}$$Let $ a, b, c>0,abc\geq 8 .$ Prove that $$\dfrac{(a-1)^2}{a^2+2}+\dfrac{(b-1)^2}{b^2+2}+\dfrac{(c-1)^2}{c^2+2}\geq\frac{25}{38}$$
This post has been edited 1 time. Last edited by sqing, Apr 16, 2025, 3:27 PM
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Strangett
2 posts
#10 • 1 Y
Y by Nuran2010
Problem 3.4. Given that \( a + b + c = 3 \), prove that
\[
(2a^2 + 3)(2b^2 + 3)(2c^2 + 3) \ge 125.
\]
Click to reveal hidden text
This post has been edited 1 time. Last edited by Strangett, Apr 16, 2025, 5:18 PM
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Novmath
20 posts
#12 • 1 Y
Y by Strangett
Strangett wrote:
Problem 3.4. Given that \( a + b + c = 3 \), prove that
\[
(2a^2 + 3)(2b^2 + 3)(2c^2 + 3) \ge 125.
\]
Click to reveal hidden text

Nice solution ,can you explain how did you think of taking f(x)= something with ln. When do we take f(x) as ln(something) ?
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Novmath
20 posts
#13
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va2010 wrote:
So a while ago, I compiled some tricks on inequalities. You are welcome to post solutions below!

The problem 2.1 (USAMO 2001 P3) is wrong. The condition was $a^2+b^2+c^2+ abc =4$
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Strangett
2 posts
#14
Y by
Novmath wrote:
Strangett wrote:
Problem 3.4. Given that \( a + b + c = 3 \), prove that
\[
(2a^2 + 3)(2b^2 + 3)(2c^2 + 3) \ge 125.
\]
Click to reveal hidden text

Nice solution ,can you explain how did you think of taking f(x)= something with ln. When do we take f(x) as ln(something) ?

Just in order to use the sum property of the natural logarithm (to apply Jensen's inequality to sums of function values). Also when taking the derivatives it is pretty obvious we are going to get something with its denominator squared and it is surely a convex function.
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sqing
42105 posts
#15
Y by
Novmath wrote:
va2010 wrote:
So a while ago, I compiled some tricks on inequalities. You are welcome to post solutions below!

The problem 2.1 (USAMO 2001 P3) is wrong. The condition was $a^2+b^2+c^2+ abc =4$
Yeah:
Let $a, b, c \geq 0$ and satisfy $ a^2+b^2+c^2 +abc = 4 . $ Show that\[ 0 \le ab + bc + ca - abc \leq 2. \]
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sqing
42105 posts
#16
Y by
Let $a, b, c \geq 0$ and $ a^2+b^2+c^2 +2abc = 4 . $ Show that$$ 0 \le ab + bc + ca - abc \leq 2$$
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Novmath
20 posts
#17
Y by
Strangett wrote:
Just in order to use the sum property of the natural logarithm (to apply Jensen's inequality to sums of function values). Also when taking the derivatives it is pretty obvious we are going to get something with its denominator squared and it is surely a convex function.
Thank you for explanation.
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Novmath
20 posts
#18
Y by
sqing wrote:
Novmath wrote:
va2010 wrote:
So a while ago, I compiled some tricks on inequalities. You are welcome to post solutions below!

The problem 2.1 (USAMO 2001 P3) is wrong. The condition was $a^2+b^2+c^2+ abc =4$
Yeah:
Let $a, b, c \geq 0$ and satisfy $ a^2+b^2+c^2 +abc = 4 . $ Show that\[ 0 \le ab + bc + ca - abc \leq 2. \]

Among a, b, c there must exist 2 numbers, say a and b, such that $ (1-a)(1-b) \geq 0$
Then: $ 1 \geq a + b - ab$
On the other hand, from $ a^2 + b^2 + c^2 + abc = 4$ we have:
$ c = \frac {-ab + \sqrt{a^2b^2 - 4a^2 - 4b^2 + 16}}{2} \leq \frac {-ab + \sqrt{a^2b^2 - 8ab + 16}}{2} = -ab +2$
Thus:
$ ab + bc + ca - abc = ab + c(a+b-ab) \leq ab + c \leq 2  (q.e.d)$
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Novmath
20 posts
#19
Y by
sqing wrote:
If $ a,b,c$ are reals show that:
$$ \frac{(b+c-a)^2}{(b+c)^2+a^2}+\frac{(c+a-b)^2}{(c+a)^2+b^2}+\frac{(a+b-c)^2}{(a+b)^2+c^2}\geq\frac{3}{5}$$

Can you share solution of this one:
Problem: Prove that for all real numbers \( a, b, c \), the following inequality holds:
\[
4(1 + a^2)(1 + b^2)(1 + c^2) \geq 3(a + b + c)^2
\]
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sqing
42105 posts
#20
Y by
Novmath wrote:
sqing wrote:
If $ a,b,c$ are reals show that:
$$ \frac{(b+c-a)^2}{(b+c)^2+a^2}+\frac{(c+a-b)^2}{(c+a)^2+b^2}+\frac{(a+b-c)^2}{(a+b)^2+c^2}\geq\frac{3}{5}$$

Can you share solution of this one:
Problem: Prove that for all real numbers \( a, b, c \), the following inequality holds:
\[
4(1 + a^2)(1 + b^2)(1 + c^2) \geq 3(a + b + c)^2
\]
https://artofproblemsolving.com/community/c6h1413227p7963638
https://artofproblemsolving.com/community/c6h1228946p6197808
https://artofproblemsolving.com/community/c6h2239135p17158474
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Novmath
20 posts
#21
Y by
sqing wrote:
Novmath wrote:
sqing wrote:
If $ a,b,c$ are reals show that:
$$ \frac{(b+c-a)^2}{(b+c)^2+a^2}+\frac{(c+a-b)^2}{(c+a)^2+b^2}+\frac{(a+b-c)^2}{(a+b)^2+c^2}\geq\frac{3}{5}$$

Can you share solution of this one:
Problem: Prove that for all real numbers \( a, b, c \), the following inequality holds:
\[
4(1 + a^2)(1 + b^2)(1 + c^2) \geq 3(a + b + c)^2
\]
https://artofproblemsolving.com/community/c6h1413227p7963638
https://artofproblemsolving.com/community/c6h1228946p6197808
https://artofproblemsolving.com/community/c6h2239135p17158474

Thank you very much :coolspeak:
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Novmath
20 posts
#22
Y by
sqing wrote:
Novmath wrote:
sqing wrote:
If $ a,b,c$ are reals show that:
$$ \frac{(b+c-a)^2}{(b+c)^2+a^2}+\frac{(c+a-b)^2}{(c+a)^2+b^2}+\frac{(a+b-c)^2}{(a+b)^2+c^2}\geq\frac{3}{5}$$

Can you share solution of this one:
Problem: Prove that for all real numbers \( a, b, c \), the following inequality holds:
\[
4(1 + a^2)(1 + b^2)(1 + c^2) \geq 3(a + b + c)^2
\]
https://artofproblemsolving.com/community/c6h1413227p7963638
https://artofproblemsolving.com/community/c6h1228946p6197808
https://artofproblemsolving.com/community/c6h2239135p17158474

Bro where can I learn separation trick in inequalities ,do you have some source which explaines it detailed, if you send it to me it will be appreciated!
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Novmath
20 posts
#23
Y by
sqing wrote:
Let $a, b, c \geq 0$ and $ a^2+b^2+c^2 +2abc = 4 . $ Show that$$ 0 \le ab + bc + ca - abc \leq 2$$

Can you also share solution of this problem:
(Vasc) Prove that if \( a, b, c \geq 0 \) and
\[
x = a + \frac{1}{b}, \quad y = b + \frac{1}{c}, \quad z = c + \frac{1}{a},
\]then
\[
xy + yz + zx \geq 2 + x + y + z.
\]
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Primeniyazidayi
98 posts
#24
Y by
Solution 3.3
This post has been edited 1 time. Last edited by Primeniyazidayi, Apr 21, 2025, 8:34 PM
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Novmath
20 posts
#25
Y by
sqing wrote:
Novmath wrote:
sqing wrote:
If $ a,b,c$ are reals show that:
$$ \frac{(b+c-a)^2}{(b+c)^2+a^2}+\frac{(c+a-b)^2}{(c+a)^2+b^2}+\frac{(a+b-c)^2}{(a+b)^2+c^2}\geq\frac{3}{5}$$

Can you share solution of this one:
Problem: Prove that for all real numbers \( a, b, c \), the following inequality holds:
\[
4(1 + a^2)(1 + b^2)(1 + c^2) \geq 3(a + b + c)^2
\]
https://artofproblemsolving.com/community/c6h1413227p7963638
https://artofproblemsolving.com/community/c6h1228946p6197808
https://artofproblemsolving.com/community/c6h2239135p17158474

Please share the solution of Vasc one that I shared
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