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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
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Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

Our full course list for upcoming classes is below:
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0 replies
jwelsh
Jul 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
Inspired by old results
sqing   0
2 minutes ago
Source: Own
Let $ a,b> 0, a^2+b^2+ab=3 .$ Prove that
$$ \frac {8} {3} \geq(a+b)^2(\frac {a} {b^2+a+1}+\frac {b} {a^2+b+1}) \geq  \frac {3(3-\sqrt 3)} {2} $$
0 replies
1 viewing
sqing
2 minutes ago
0 replies
Interesting geometrical configuration
SeboS   1
N 4 minutes ago by Lil_flip38
Let $ABC$ be a triangle with orthocentre $H$, and altitudes $D,E,F$ on sides $(BC), (AC), (AB)$
If $$ (HBC) \cap (HEF)=X $$$$(AHC) \cap (DHF)=Y $$$$(ABH) \cap (DEH)=Z $$Prove that the points $ H, X, Y, Z $ are cyclic
1 reply
SeboS
12 minutes ago
Lil_flip38
4 minutes ago
4 var inequality
ehuseyinyigit   2
N 8 minutes ago by ehuseyinyigit
Source: Own
Let $a,b,c,d$ be positive real numbers. Prove
$$\sum{a^2}+3\sum{a^2b^2c^2}+\sum{ab^2c}+12abcd$$$$\geq 2abcd[(a+c)(b+d)-ac-bd]+3\sum{a^2bc}+\sum{abc^2}$$
2 replies
1 viewing
ehuseyinyigit
Yesterday at 2:53 PM
ehuseyinyigit
8 minutes ago
Interesting functional equation
TheUltimate123   14
N 9 minutes ago by math-olympiad-clown
Source: ELMO Shortlist 2023 A2
Let \(\mathbb R_{>0}\) denote the set of positive real numbers. Find all functions \(f:\mathbb R_{>0}\to\mathbb R_{>0}\) such that for all positive real numbers \(x\) and \(y\), \[f(xy+1)=f(x)f\left(\frac1x+f\left(\frac1y\right)\right).\]
Proposed by Luke Robitaille
14 replies
TheUltimate123
Jun 29, 2023
math-olympiad-clown
9 minutes ago
A specific case of my previous conjecture
Rhapsodies_pro   1
N 24 minutes ago by nexu
Source: n=4
Prove that \(3\) is the largest value of the constant \(k\) such that \[{ab+ac+ad+bc+bd+cd-6}\leqslant{k{\left(a+b+c+d-1\right)}{\left(a+b+c+d-4\right)}}\]holds for any nonnegative real numbers \(a, b, c, d\) satisfying \({a^2+b^2+c^2+d^2+5abcd}\geqslant9\).
1 reply
Rhapsodies_pro
Wednesday at 4:38 PM
nexu
24 minutes ago
3 var inequality
ehuseyinyigit   9
N 25 minutes ago by nexu
Source: Own
Let $x,y,z$ be positive real numbers. Prove that

$$\dfrac{x^3+72xy^2}{z^3+x^2y}+\dfrac{y^3+72yz^2}{x^3+y^2z}+\dfrac{z^3+72zx^2}{y^3+z^2x}\geq \dfrac{15}{2}+\dfrac{102xyz(x+y+z)}{x^3y+y^3z+z^3x}$$
9 replies
ehuseyinyigit
Jul 21, 2025
nexu
25 minutes ago
Four variables
Nguyenhuyen_AG   3
N 32 minutes ago by nexu
Let $a,\,b,\,c,\,d$ non-negative real numbers. Prove that
\[\frac{abc}{(a+b+c)^3}+\frac{bcd}{(b+c+d)^3}+\frac{cda}{(c+d+a)^3}+\frac{dab}{(d+a+b)^3} \leqslant \frac{(a+b+c+d)^2}{27(a^2+b^2+c^2+d^2)}.\]
3 replies
Nguyenhuyen_AG
Jul 23, 2025
nexu
32 minutes ago
Inequality
SunnyEvan   5
N 33 minutes ago by nexu
Source: Own
Let $ a,b,c >0$, such that: $ a+b+c=3 .$ Prove that:
$$ 2025 \geq (628-96(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}))(ab+bc+ca)+864\frac{a^3+b^3+c^3}{a^2+b^2+c^2}$$When does the equality hold ?
5 replies
SunnyEvan
Jul 18, 2025
nexu
33 minutes ago
The refinement of GMA 567
mihaig   6
N 37 minutes ago by mihaig
Source: Own
Let $a_1,\ldots, a_{n}\geq0~~(n\geq4)$ be real numbers such that
$$\sum_{i=1}^{n}{a_i^2}+(n^2-3n+1)\prod_{i=1}^{n}{a_i}\geq(n-1)^2.$$Prove
$$\left(\sum_{i=1}^{n}{a_i}\right)^2+\frac{2n-1}{(n-1)^3}\cdot\sum_{1\leq i<j\leq n}{\left(a_i-a_j\right)^2}\geq n^2.$$
6 replies
mihaig
Yesterday at 11:22 AM
mihaig
37 minutes ago
IMO ShortList 1999, number theory problem 1
orl   65
N 42 minutes ago by SSS_123
Source: IMO ShortList 1999, number theory problem 1
Find all the pairs of positive integers $(x,p)$ such that p is a prime, $x \leq 2p$ and $x^{p-1}$ is a divisor of $ (p-1)^{x}+1$.
65 replies
orl
Nov 13, 2004
SSS_123
42 minutes ago
Tough with weird constraint
mihaig   1
N 42 minutes ago by nexu
Source: Own
Let $a,b,c,d\geq0$ satisfying
$$\left(a+b+c+d\right)^2+\frac7{27}\cdot\sum_{\text{sym}}{\left(a-b\right)^2}=16.$$Prove
$$a^2+b^2+c^2+d^2+5abcd\leq9.$$
1 reply
mihaig
Today at 4:08 AM
nexu
42 minutes ago
Four variables (3)
Nguyenhuyen_AG   2
N 44 minutes ago by nexu
Let $a,\,b,\,c,\,d$ be non-negative real numbers. Prove that
\[\frac{24a^3 + 49bcd}{(b + c + d)^3} + \frac{24b^3+49cda }{(c + d + a)^3} + \frac{24c^3+49dab}{(d + a + b)^3} + \frac{24d^3+49abc}{(a + b + c)^3} \geqslant \frac{292}{27}.\]
2 replies
Nguyenhuyen_AG
Yesterday at 9:25 AM
nexu
44 minutes ago
5-var inequality
sqing   3
N an hour ago by nexu
Source: Zhaobin
Let $ a,b,c,d,e>0 . $ Prove that$$\frac {a^2}{b} +\frac {b^2}{c} +\frac {c^2}{d} +\frac {d^2}{e} +\frac {e^2}{a}  \geq \sqrt{5(a^2+b^2+c^2+d^2+e^2)}$$
3 replies
sqing
Jun 18, 2025
nexu
an hour ago
Integer-Valued FE comes again
lminsl   217
N an hour ago by TigerOnion
Source: IMO 2019 Problem 1
Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $a$ and $b$, $$f(2a)+2f(b)=f(f(a+b)).$$Proposed by Liam Baker, South Africa
217 replies
lminsl
Jul 16, 2019
TigerOnion
an hour ago
Function Equation
MathsII-enjoy   5
N Jun 24, 2025 by EvansGressfield
Find all function $f: \mathbb{R^+} \to \mathbb{R^+}$ such that for all $x, y\in\mathbb{R^+}$: $$f(2y+f(xy))+2x=f(x).f(y)$$
5 replies
MathsII-enjoy
Jun 24, 2025
EvansGressfield
Jun 24, 2025
Function Equation
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MathsII-enjoy
105 posts
#1 • 1 Y
Y by pomodor_ap
Find all function $f: \mathbb{R^+} \to \mathbb{R^+}$ such that for all $x, y\in\mathbb{R^+}$: $$f(2y+f(xy))+2x=f(x).f(y)$$
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youochange
200 posts
#2
Y by
Can someone pls verify my sol!
ah fakesolve
This post has been edited 1 time. Last edited by youochange, Jun 24, 2025, 2:16 PM
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GreekIdiot
323 posts
#3 • 1 Y
Y by youochange
Why must $f$ be linear?
Plus note that you havent proven $f(u)-f(v)=u-v$ for all positive reals.
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rrrMath
75 posts
#4
Y by
Denote the given assertion by $P\left(x,y\right)$.
$P\left(x,1\right)$:$$f\left(2+f\left(x\right)\right)+2x=f\left(x\right)f\left(1\right)$$Thus f injective.
For fixed $a>0$ and $P\left(\frac{2}{a},\frac{a}{2}y\right)$ we obtain $f\left(y\right)+ay$ injective thus f increasing.
Denote:
$$c=\lim_{x\to0^+}{f\left(x\right)},d=\lim_{x\to c^+}{f\left(x\right)}$$And taking $P\left(x,y\to0^+\right)$:
$$d+2x=cf\left(x\right)$$Thus f linear, checking $ax+b$ gives:
$$a^2xy+2ay+2x+ab+b=a^2xy+abx+aby+b^2$$Thus only solution is $\forall x\in\mathbb{R}^+:f\left(x\right)=x+2$ which clearly fits.
This post has been edited 3 times. Last edited by rrrMath, Jun 24, 2025, 2:04 PM
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youochange
200 posts
#5
Y by
GreekIdiot wrote:
Why must $f$ be linear?
Plus note that you havent proven $f(u)-f(v)=u-v$ for all positive reals.

Thanks sir
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EvansGressfield
6 posts
#6
Y by
https://artofproblemsolving.com/community/q1h3397330p32677734
This function can be solved similarly to IMOSL 2020
This post has been edited 1 time. Last edited by EvansGressfield, Jun 24, 2025, 2:56 PM
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