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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
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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
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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
1 viewing
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
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Inequality
SunnyEvan   4
N a few seconds ago by Jackson0423
Let $ a,b,c \in R $ such that: $ abc>0 $ and $a^2+b^2+c^2=4(ab+bc+ca)$
Prove that: $$\frac{abc(a^3+b^3+c^3+ab(a+b)+bc(b+c)+ca(c+a))}{730a^2b^2c^2+k((a-b)(b-c)(c-a))^2} \leq \frac{2}{\sqrt{\frac{730}{3}k-9k^2}-3k} $$Where $ k\in(0,\frac{365(2-\sqrt2)}{54}]. $


$$ \frac{abc(a^3+b^3+c^3+ab(a+b)+bc(b+c)+ca(c+a))}{730a^2b^2c^2+k((a-b)(b-c)(c-a))^2} \leq \frac{18(\sqrt2+1)}{365} $$Where $ k\in[\frac{365(2-\sqrt2)}{54}, +\infty). $


$$ \frac{abc(a^3+b^3+c^3+ab(a+b)+bc(b+c)+ca(c+a))}{730a^2b^2c^2+k((a-b)(b-c)(c-a))^2} \geq \frac{18(\sqrt2+1)}{365} $$Where $ k\in(-\infty ,0). $
4 replies
SunnyEvan
Jul 1, 2025
Jackson0423
a few seconds ago
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Geometry
Invertibility   1
N 2 minutes ago by MathLuis
Source: Own (Switzerland Slovenia Liechtenstein Revenge exam 2025)
Let $ABC$ be a triangle with circumcircle $\Omega$ and circumcenter $O$. Let $CO\cap{}AB = D$ and $BO\cap{}AC = E$. Define $X$ as the midpoint of $DE$ and let the second intersections of rays $CX$ and $BX$ with $\Omega$ be $N$ and $M$, respectively. Finally, define $P$ as the intersection of the lines $ND$ and $ME$.

Prove that $BC$ passes through the circumcenter of $AOP$.
1 reply
Invertibility
Jul 12, 2025
MathLuis
2 minutes ago
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3 Var (?)
SunnyEvan   5
N 2 minutes ago by SunnyEvan
Source: Own
Let $ a,b,c \in R^+ $, such that :$ab+bc+ca+abc=2$.
Prove that: $$ \frac{1}{3\sqrt{a^2(b+1)(c+1)+abc+2}}+\frac{1}{3\sqrt{b^2(c+1)(a+1)+abc+2}}+\frac{1}{5\sqrt{c^2(a+1)(b+1)+abc+2}} < \frac{43}{90} $$$$ \frac{1}{3\sqrt{a^2(b+1)(c+1)+abc+2}}+\frac{1}{4\sqrt{b^2(c+1)(a+1)+abc+2}}+\frac{1}{5\sqrt{c^2(a+1)(b+1)+abc+2}} < \frac{5}{12} $$
5 replies
SunnyEvan
Jul 18, 2025
SunnyEvan
2 minutes ago
J
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fun problem
Laan   1
N 5 minutes ago by genius_007
proof that the convex polygon that has the most right angles is a rectangle (4 right-angles)
1 reply
Laan
an hour ago
genius_007
5 minutes ago
J
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2025 IMO Results
ilikemath247365   11
N 7 minutes ago by ilikemath247365
Source: https://www.imo-official.org/year_info.aspx?year=2025
Congrats to China for getting 1st place! Congrats to USA for getting 2nd and congrats to South Korea for getting 3rd!
11 replies
ilikemath247365
Yesterday at 4:48 PM
ilikemath247365
7 minutes ago
J
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functional equation reals
danepale   36
N 23 minutes ago by Sqrt2_1.4142
Source: Croatia TST 2016
Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that for all real $x,y$:
$$ f(x^2) + xf(y) = f(x) f(x + f(y)) \, . $$
36 replies
danepale
Apr 25, 2016
Sqrt2_1.4142
23 minutes ago
J
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Strange symmetric multi-equation
giangtruong13   1
N 34 minutes ago by iniffur
Solve the multi-equation: $\begin{cases} 5x(y^2-1)=4(x^2+y^2) \\ 5y(x^2+1)=3(x^2+y^2) \end{cases}$
1 reply
giangtruong13
Yesterday at 3:13 PM
iniffur
34 minutes ago
J
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AC, BF, DE concurrent
a1267ab   78
N 44 minutes ago by TigerOnion
Source: APMO 2020 Problem 1
Let $\Gamma$ be the circumcircle of $\triangle ABC$. Let $D$ be a point on the side $BC$. The tangent to $\Gamma$ at $A$ intersects the parallel line to $BA$ through $D$ at point $E$. The segment $CE$ intersects $\Gamma$ again at $F$. Suppose $B$, $D$, $F$, $E$ are concyclic. Prove that $AC$, $BF$, $DE$ are concurrent.
78 replies
a1267ab
Jun 9, 2020
TigerOnion
44 minutes ago
J
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Find the number
Thanhdoan1   1
N an hour ago by CuriousMathBoy72
Find all the positive number x such that x^3+3^x is a square.
1 reply
1 viewing
Thanhdoan1
an hour ago
CuriousMathBoy72
an hour ago
J
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The refinement of GMA 567
mihaig   2
N an hour 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
$$\left(\sum_{i=1}^{n}{a_i}\right)^2+\frac{2n-1}{(n-1)^3}\cdot\sum_{1\leq i<j\leq n}{\left(a_i-a_j\right)^2}\geq n^2.$$
2 replies
mihaig
5 hours ago
mihaig
an hour ago
J
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sum x ^ 3/ (x + y) >= (xy + yz + zx)/2
parmenides51   4
N an hour ago by SunnyEvan
Source: 2003 Kazakhstan MO grade XI P2
For positive real numbers $ x, y, z $, prove the inequality: $$ \displaylines {\frac {x ^ 3} {x + y} + \frac {y ^ 3} {y + z} + \frac {z ^ 3} {z + x} \geq \frac {xy + yz + zx} {2}.} $$
4 replies
parmenides51
Nov 3, 2020
SunnyEvan
an hour ago
J
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Inspired by old results
sqing   3
N an hour ago by Dr1001Sa
Source: Own
Let $ x,y\geq 0, x+y=2. $ Prove that$$ \sqrt{x^2+6y+2}+\sqrt{y^2+6x+2}+\sqrt{xy+8}\geq 9$$$$ \sqrt{x^2+7y+1}+\sqrt{y^2+7x+1}+\sqrt{xy+8}\geq \sqrt{2\left(14+2\sqrt{30}+\sqrt{5(23+4\sqrt{30})}\right)}$$
3 replies
sqing
Jul 9, 2025
Dr1001Sa
an hour ago
J
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Useless identity
mashumaro   2
N an hour ago by AbhayAttarde01
Source: Own
Let $a_1$, $a_2$, $\dots$, $a_6$ be reals, and let $f(m, n) = \sum_{i=m}^{n} a_i$. Show that
\[ f(5,5)f(5,6) = f(2,4)f(3,4) = f(1,4)f(4,4) \implies f(1,1)f(1,2)f(4,5)f(4,6) = f(2,3)f(3,3)f(1,5)f(1,6) \]
2 replies
mashumaro
Yesterday at 6:01 AM
AbhayAttarde01
an hour ago
J
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Seems Complicated but Trivial after inversion over T
zqy648   0
an hour ago
Source: 2025 New Year 谜之竞赛-2
In \(\triangle ABC\), let \(I\) be the incenter and \(\omega\) be the incircle. \(\omega\) is tangent to \(BC\) at $D$, and let \(T\) be the tangent point of \(A\)-pseudo-incircle and \(\odot( ABC)\). A circle centered at \(T\) with radius \(TI\) intersects \(\omega\) at \(X\) and \(Y\).

Prove that the circumcircle of \(\triangle AXY\) is tangent to the circumcircles of \(\triangle ABC\), \(\triangle BTD\), and \(\triangle CTD\).

Proposed by Jiaye Cai
0 replies
zqy648
an hour ago
0 replies
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Polynomial method of moving points
MathHorse   6
N May 23, 2025 by Potyka17
Two Hungarian math olympians achieved significant breakthroughs in the field of polynomial moving points. Their main results are summarised in the attached pdf. Check it out!
6 replies
MathHorse
Jun 30, 2023
Potyka17
May 23, 2025
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Polynomial method of moving points
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Reveal topic tags
algebra
polynomial
article
projective geometry
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MathHorse
53 posts
MathHorse
#1 h  Jun 30, 2023, 12:38 PM • 5 Y
Y by SenpaiMarton, ImSh95, jatekos101, Vlados021, MathLuis
Two Hungarian math olympians achieved significant breakthroughs in the field of polynomial moving points. Their main results are summarised in the attached pdf. Check it out!
Attachments:
polynomial_moving_points.pdf (157kb)
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PureRun89
120 posts
PureRun89
#2 h  Jun 30, 2023, 1:55 PM • 1 Y
Y by ImSh95
Actually there have been the theory of moving points years ago...
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SenpaiMarton
5 posts
SenpaiMarton
#3 h  Jun 30, 2023, 2:31 PM • 1 Y
Y by ImSh95
Though the technique is well-known, some lemmas and theorems in the attachment are hard to find online. Perhaps a breakthrough would be an overstatement, but I still find it rather useful to have this as a resource.
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Laszlo_but_not_Lovasz
2 posts
Laszlo_but_not_Lovasz
#4 h  Jun 30, 2023, 6:46 PM
Y by
Yeah, I also think it's nice to have an easily accessible source of Moving Points on AoPS, and it can also be useful for some IMO contestants.
This post has been edited 1 time. Last edited by Laszlo_but_not_Lovasz, Jun 30, 2023, 6:47 PM
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JamiroQuai
1 post
JamiroQuai
#5 h  Jul 7, 2023, 11:20 AM
Y by
This is the first useful extension of the method that I found since the results outlined by Vladyslav Zveryk in 2019. I think these techniques will be mainstream in a couple of years.
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Potyka17
3 posts
Potyka17
#6 h  Jul 7, 2023, 4:03 PM
Y by
The original, and more extended hungarian version. Written in hungarian, sorry :(( Just putting it here so that it can be used on IMO
https://drive.google.com/file/d/1q04cNsBx3DXzdeoZzAuyOZQXIW8n6GgR/view?usp=drive_link
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Potyka17
3 posts
Potyka17
#7 h  May 23, 2025, 3:15 PM
Y by
Precise proofs of the method can be found here (in english): https://drive.google.com/file/d/1hMWDUDInet4nwoecKWS-_2PuKPFtsR9Z/view?usp=sharing
Excercises to fully learn the method (in hungarian): https://drive.google.com/file/d/1LwjI1nk4vd-TpauP2_HPn85qU2fTN1ti/view?usp=sharing
This post has been edited 1 time. Last edited by Potyka17, May 23, 2025, 3:16 PM
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