Plan ahead for the next school year. Schedule your class today!

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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
and train with the best! Please note that early bird pricing ends August 19th!
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
Find the number
Thanhdoan1   3
N a minute ago by CuriousMathBoy72
Find all the positive number x such that x^3+3^x is a square.
3 replies
Thanhdoan1
Yesterday at 2:39 PM
CuriousMathBoy72
a minute ago
FE : f(x+f(y))+f(y+f(x))=f(f(2x)+2y)
Valjeanpi   3
N 14 minutes ago by Bridgeon
Find all surjective fonctions $f:\mathbb R ^+ \backslash\{0\}\rightarrow \mathbb R ^+ \backslash\{0\}$ verifying :
$$f(x+f(y))+f(y+f(x))=f(f(2x)+2y)$$for all $x,y\in \mathbb R ^+ \backslash\{0\} $
3 replies
Valjeanpi
Feb 15, 2023
Bridgeon
14 minutes ago
computational, cyclic ABCD
parmenides51   1
N 15 minutes ago by SuperBarsh
Source: Mexican Geometry Olympiad 2014 (OMG) - La Geometrense p2 https://artofproblemsolving.com/community/c2664039_2014_mexican_geometr
Quadrilateral $ABCD$ is inscribed in a circle of radius $ 1$, such that the diagonal $AC$ is a diameter and $BD=AB$. Diagonals intersect at $P$. It is known that $PC=\frac25$ . What is the length of side $CD$?
1 reply
parmenides51
Nov 28, 2021
SuperBarsh
15 minutes ago
orang NT
KevinYang2.71   26
N 19 minutes ago by blug
Source: ISL 2024 N1
Find all positive integers $n$ with the following property: for all positive divisors $d$ of $n$, we have $d+1\mid n$ or $d+1$ is prime.
26 replies
KevinYang2.71
Jul 16, 2025
blug
19 minutes ago
Combinatorics
jawadkaleem   2
N 19 minutes ago by Alphabeta123
Michel starts with the string HMMT.
An operation consists of either replacing an occurrence of H with HM, replacing an
occurrence of MM with MOM, or replacing an occurrence of T with MT. For example,
the two strings that can be reached after one operation are HMMMT and HMOMT.
Compute the number of distinct strings Michel can obtain after exactly 10 operation
2 replies
jawadkaleem
Jul 2, 2025
Alphabeta123
19 minutes ago
An easy geometry in Taiwan TST
Li4   8
N 25 minutes ago by Aiden-1089
Source: 2022 Taiwan TST Round 3 Independent Study 1-G
Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Omega$. Let $M$ be the midpoint of side $BC$. Point $D$ is chosen from the minor arc $BC$ on $\Gamma$ such that $\angle BAD = \angle MAC$. Let $E$ be a point on $\Gamma$ such that $DE$ is perpendicular to $AM$, and $F$ be a point on line $BC$ such that $DF$ is perpendicular to $BC$. Lines $HF$ and $AM$ intersect at point $N$, and point $R$ is the reflection point of $H$ with respect to $N$.

Prove that $\angle AER + \angle DFR = 180^\circ$.

Proposed by Li4.
8 replies
Li4
Apr 27, 2022
Aiden-1089
25 minutes ago
Inspired by old results
sqing   1
N 29 minutes ago by sqing
Source: Own
Let $ a,b> 0, a^2+b^2+ab=3 .$ Prove that
$$ 3\sqrt 3 \geq(a+b)^2(\frac {a} {b^2+b+1}+\frac {b} {a^2+a+1}) \geq   \frac {8} {3}$$$$ \frac {8} {3} \geq(a+b)^2(\frac {a} {b^2+a+1}+\frac {b} {a^2+b+1}) \geq  \frac {3(3-\sqrt 3)} {2} $$
1 reply
sqing
39 minutes ago
sqing
29 minutes ago
6 points concyclic wanted, touchpoints of incircle related
parmenides51   1
N 32 minutes ago by SuperBarsh
Source: Mexican Geometry Olympiad 2014 (OMG) - La Geometrense p1 https://artofproblemsolving.com/community/c2664039_2014_mexican_geometr
The incircle of triangle $ABC$ touches sides $BC$, $CA$, and $AB$ at $A'$, $B'$, and $C'$ respectively. $G$ is the intersection point of $AA '$, $BB'$ and $CC '$. The circumcircle of $GA'B'$ again touches $AC$ and $BC$ at points $C_A$ and $C_B$ respectively. Similarly are defined points $B_A$, $B_C$, $A_B$ and $A_C$ . Show that the points $C_A$, $C_B$, $B_A$, $B_C$, $A_B$ and $A_C$ all lie on the same circle.
1 reply
parmenides51
Nov 28, 2021
SuperBarsh
32 minutes ago
Interesting geometrical configuration
SeboS   1
N 42 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
an hour ago
Lil_flip38
42 minutes ago
4 var inequality
ehuseyinyigit   2
N an hour 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
an hour ago
Interesting functional equation
TheUltimate123   14
N an hour 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
an hour ago
A specific case of my previous conjecture
Rhapsodies_pro   1
N an hour 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
an hour ago
3 var inequality
ehuseyinyigit   9
N an hour 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
an hour ago
Four variables
Nguyenhuyen_AG   3
N an hour 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
an hour ago
Combinatorial Sum
P162008   0
Apr 24, 2025
$\frac{\sum_{r=0}^{24} \binom{100}{4r} \binom{100}{4r + 2}}{\sum_{r=1}^{25} \binom{200}{8r - 6}}$ is equal to
0 replies
P162008
Apr 24, 2025
0 replies
Combinatorial Sum
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P162008
262 posts
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$\frac{\sum_{r=0}^{24} \binom{100}{4r} \binom{100}{4r + 2}}{\sum_{r=1}^{25} \binom{200}{8r - 6}}$ is equal to
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