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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
J
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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
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
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Integer-Valued FE comes again
lminsl   215
N 4 minutes ago by zhoujef000
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
215 replies
+1 w
lminsl
Jul 16, 2019
zhoujef000
4 minutes ago
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Japan 1997 inequality
hxtung   81
N 6 minutes ago by zhoujef000
Source: Japan MO 1997, problem #2
Prove that

$ \frac{\left(b+c-a\right)^{2}}{\left(b+c\right)^{2}+a^{2}}+\frac{\left(c+a-b\right)^{2}}{\left(c+a\right)^{2}+b^{2}}+\frac{\left(a+b-c\right)^{2}}{\left(a+b\right)^{2}+c^{2}}\geq\frac35$

for any positive real numbers $ a$, $ b$, $ c$.
81 replies
1 viewing
hxtung
Jul 27, 2003
zhoujef000
6 minutes ago
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Inequality with xyz>=1
Doom   2
N 12 minutes ago by zhoujef000
Let $x, y, z$ be positive reals such that $xyz\geq 1$. Prove that \[\frac{x}{x^3+y^2+z}+\frac{y}{y^3+z^2+x}+\frac{z}{z^3+x^2+y}\leq 1 \text{ .}\]
2 replies
Doom
Aug 29, 2013
zhoujef000
12 minutes ago
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Polynomial of Degree n
Brut3Forc3   22
N 17 minutes ago by zhoujef000
Source: 1975 USAMO Problem 3
If $ P(x)$ denotes a polynomial of degree $ n$ such that $ P(k)=\frac{k}{k+1}$ for $ k=0,1,2,\ldots,n$, determine $ P(n+1)$.
22 replies
Brut3Forc3
Mar 15, 2010
zhoujef000
17 minutes ago
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inequality, combinatorics
Ducksohappi   2
N 20 minutes ago by Ducksohappi
Let $x_i, i =1,2,...,n$ be non-negative integers such that:
$\sum_{i=1}^n x_i=k$. find the minimum value of $T=\sum_{i=1}^n \binom{x_i}{7}$
(assume that $\binom{a}{b}=0$ if $a<b$)
2 replies
Ducksohappi
Yesterday at 11:22 AM
Ducksohappi
20 minutes ago
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Inspired by Michael Rozenberg
sqing   1
N 24 minutes ago by sqing
Source: Own
Let $a,b,c>0. $ Prove that
$$(a + b - c)\left( \frac{2}{a + b} - \frac{1}{b + c} - \frac{1}{c + a} \right) \leq5 -2\sqrt 6$$$$(a + b - 2c)\left( \frac{3}{a + b} - \frac{1}{b + c} - \frac{1}{c + a} \right) \leq 10-4\sqrt 6$$h h
1 reply
sqing
41 minutes ago
sqing
24 minutes ago
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Functional Equation
AnhQuang_67   6
N an hour ago by jasperE3
Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $$2\cdot f\Big(\dfrac{-xy}{2}+f(x+y)\Big)=xf(y)+yf(x), \forall x, y \in \mathbb{R} $$
6 replies
AnhQuang_67
Apr 4, 2025
jasperE3
an hour ago
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Inspired by old results
sqing   1
N an hour ago by sqing
Source: Own
Let $ a,b,c\geq 0 $ and $ ab+bc+ca+a^2+b^2+c^2=6. $ Prove that
$$a^4+b^4+c^4+5abc\geq 8$$$$a^3+b^3+c^3+\frac{332}{125}abc\geq \frac{707}{125}$$$$a^3+b^3+c^3+\frac{133}{50}abc\geq 4\sqrt{2}$$$$a^4+b^4+c^4+\frac{124}{25}abc\geq \frac{199}{25}$$
1 reply
sqing
2 hours ago
sqing
an hour ago
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Challenging trigonometric inequality
VoidMutsumi   1
N 2 hours ago by VoidMutsumi
(I) Let $p,q$ are coprime positive odd inetegers, prove the inequality:
$$ pq(1-|p-q|) \leqslant \sum_{m=0}^{p-1} \sum_{n=0}^{q-1} \frac{2}{\cos(\frac{2m\pi}{p}) + \cos(\frac{2n\pi}{q})} \leqslant pq(1+|p-q|). $$(II) An odd number $q>1$ is called right-good (or left-good), if there exists an odd integer $p\in[1,q)$, coprime to $q$, for which the right-hand (or left-hand) inequality above holds with equality. Prove the following:
(a) Every odd integer $q>1$ is right-good;
(b) An odd integer $q>1$ is left-good if and only if $q+1$ is not a power of 2.
1 reply
VoidMutsumi
2 hours ago
VoidMutsumi
2 hours ago
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Powers of 2 - Iran NMO 1999 (Second Round) Problem1
sororak   6
N 2 hours ago by mudkip42
Does there exist a positive integer that is a power of $2$ and we get another power of $2$ by swapping its digits? Justify your answer.
6 replies
sororak
Oct 4, 2010
mudkip42
2 hours ago
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powers of 2 permutations?
megahertz13   5
N 2 hours ago by mudkip42
Source: 1998 ToT
Decide whether there exist two distinct powers of $2$ which have the same digits, but in a different order. (We forbid leading zeros.)

5 replies
megahertz13
Nov 23, 2024
mudkip42
2 hours ago
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2025 IMO P2 Geometry Problem Video Solution and Extension
Plane_geometry_youtuber   0
2 hours ago
Hi,

I posted this video a few days ago as the solution and extension on this problem. Welcome to comment.

https://www.youtube.com/watch?v=0TcMSrOYZ7c&t=1207s

Best,
0 replies
1 viewing
Plane_geometry_youtuber
2 hours ago
0 replies
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IMO 2025 P2
sarjinius   87
N 2 hours ago by Plane_geometry_youtuber
Source: 2025 IMO P2
Let $\Omega$ and $\Gamma$ be circles with centres $M$ and $N$, respectively, such that the radius of $\Omega$ is less than the radius of $\Gamma$. Suppose $\Omega$ and $\Gamma$ intersect at two distinct points $A$ and $B$. Line $MN$ intersects $\Omega$ at $C$ and $\Gamma$ at $D$, so that $C, M, N, D$ lie on $MN$ in that order. Let $P$ be the circumcentre of triangle $ACD$. Line $AP$ meets $\Omega$ again at $E\neq A$ and meets $\Gamma$ again at $F\neq A$. Let $H$ be the orthocentre of triangle $PMN$.

Prove that the line through $H$ parallel to $AP$ is tangent to the circumcircle of triangle $BEF$.

Proposed by Trần Quang Hùng, Vietnam
87 replies
+1 w
sarjinius
Jul 15, 2025
Plane_geometry_youtuber
2 hours ago
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Interesting inequality
sqing   2
N 2 hours ago by sqing
Source: Own
Let $ a,b$ be real numbers such that $ a+b=2 $. Prove that
$$ka^2b^2+ (a-1)(b-1)(ab-1) \geq\frac{k}{k+1}$$Where $ k>0.$
$$a^2b^2+ (a-1)(b-1)(ab-1) \geq\frac{1}{2}$$$$2a^2b^2+ (a-1)(b-1)(ab-1) \geq\frac{2}{3}$$$$a^2b^2+ 2(a-1)(b-1)(ab-1) \geq\frac{2}{3}$$
2 replies
sqing
Yesterday at 1:02 PM
sqing
2 hours ago
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equal faces in octahedron with triangular faces , // in pairs
parmenides51   0
Oct 11, 2021
Source: 1995 Argentina IberoAmerican Training List p27 https://artofproblemsolving.com/community/c2434876_argentina_training_lists__geom
Show that if an octahedron has all its faces triangular and these faces are parallel in pairs, then the faces are equal in pairs.
0 replies
parmenides51
Oct 11, 2021
0 replies
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equal faces in octahedron with triangular faces , // in pairs
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Reveal topic tags
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Source: 1995 Argentina IberoAmerican Training List p27 https://artofproblemsolving.com/community/c2434876_argentina_training_lists__geom
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parmenides51
30655 posts
parmenides51
#1 h  Oct 11, 2021, 7:49 PM
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Show that if an octahedron has all its faces triangular and these faces are parallel in pairs, then the faces are equal in pairs.
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