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k a August Highlights and 2025 AoPS Online Class Information
jwelsh   0
Friday at 2:14 PM
CONGRATULATIONS to all the competitors at this year’s International Mathematical Olympiad (IMO)! The US Team took second place with 5 gold medals and 1 silver - we are proud to say that each member of the 2025 IMO team has participated in an AoPS WOOT (Worldwide Online Olympiad Training) class!

"As a parent, I'm deeply grateful to AoPS. Tiger has taken very few math courses outside of AoPS, except for a local Math Circle that doesn't focus on Olympiad math. AoPS has been one of the most important resources in his journey. Without AoPS, Tiger wouldn't be where he is today — especially considering he's grown up in a family with no STEM background at all."
— Doreen Dai, parent of IMO US Team Member Tiger Zhang

Interested to learn more about our WOOT programs? Check out the course page here or join a Free Scheduled Info Session. Early bird pricing ends August 19th!:
CodeWOOT Code Jam - Monday, August 11th
ChemWOOT Chemistry Jam - Wednesday, August 13th
PhysicsWOOT Physics Jam - Thursday, August 14th
MathWOOT Math Jam - Friday, August 15th

There is still time to enroll in our last wave of summer camps that start in August at the Virtual Campus, our video-based platform, for math and language arts! From Math Beasts Camp 6 (Prealgebra Prep) to AMC 10/12 Prep, you can find an informative 2-week camp before school starts. Plus, our math camps don’t have homework and cover cool enrichment topics like graph theory. Our language arts courses will build the foundation for next year’s challenges, such as Language Arts Triathlon for levels 5-6 and Academic Essay Writing for high school students.

Lastly, Fall is right around the corner! 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. We’ve opened new Academy locations in San Mateo, CA, Pasadena, CA, Saratoga, CA, Johns Creek, GA, Northbrook, IL, and Upper West Side (NYC), New York.

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0 replies
jwelsh
Friday at 2:14 PM
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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Number Theory
JetFire008   1
N 6 minutes ago by blug
Source: Elementary Number Theory by David M. Burton
Modify Euclid's proof that there are infinitely many primes by assuming the existence of a largest prime $p$ and using the integer $N=p!+1$ to arrive at a contradiction.
1 reply
JetFire008
an hour ago
blug
6 minutes ago
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Partial products from p-2 numbers
JustPostNorthKoreaTST   2
N 7 minutes ago by navid
Source: 2015 North Korea Mathematical Olympiad P2
Let $p$ be an odd prime and $a_1,a_2,\ldots,a_{p-2}$ be positive integers (not necessarily different), such that for any $k \in \{1,2,\ldots,p-2\}$, we have $p \nmid a_k$ and $p \nmid (a_k^k-1)$. Show that one can pick some numbers from $a_1,a_2,\ldots,a_{p-2}$ such that their product is $\equiv 2 \pmod p$.
2 replies
JustPostNorthKoreaTST
3 hours ago
navid
7 minutes ago
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Bonza functions
KevinYang2.71   72
N 9 minutes ago by pie854
Source: 2025 IMO P3
Let $\mathbb{N}$ denote the set of positive integers. A function $f\colon\mathbb{N}\to\mathbb{N}$ is said to be bonza if
\[ f(a)~~\text{divides}~~b^a-f(b)^{f(a)} \]for all positive integers $a$ and $b$.

Determine the smallest real constant $c$ such that $f(n)\leqslant cn$ for all bonza functions $f$ and all positive integers $n$.

Proposed by Lorenzo Sarria, Colombia
72 replies
KevinYang2.71
Jul 15, 2025
pie854
9 minutes ago
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Next term is sum of three largest proper divisors
vsamc   24
N 14 minutes ago by pie854
Source: 2025 IMO P4
A proper divisor of a positive integer $N$ is a positive divisor of $N$ other than $N$ itself.

The infinite sequence $a_1, a_2, \cdots$ consists of positive integers, each of which has at least three proper divisors. For each $n\geq 1$, the integer $a_{n+1}$ is the sum of the three largest proper divisors of $a_n$.

Determine all possible values of $a_1$.

Proposed by Paulius Aleknavičius, Lithuania
24 replies
vsamc
Jul 16, 2025
pie854
14 minutes ago
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Inspired by lgx57
sqing   1
N 26 minutes ago by sqing
Source: Own
Let $ a,b,c \in[0,1],ab+bc+ca=1. $ Prove that
$$\dfrac{37+30\sqrt{3}}{27}\leq (a^2+ 2b)(b^2+2c)(c^2+2a) \leq6$$$$\dfrac{(1+\sqrt{3})^3}{27}\leq (a^2+ b)(b^2+c)(c^2+a) \leq 2$$
1 reply
1 viewing
sqing
an hour ago
sqing
26 minutes ago
J
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Century Value
EthanWYX2009   1
N 29 minutes ago by Photaesthesia
Source: 2025 CSMC-5
For a permutation \(\pi: a_1, a_2, \cdots, a_{100}\) of \(1, 2, \cdots, 100\), define[list]
[*]The number of fixed points as \(\sigma(\pi) = \left|\{ k \mid a_k = k, \, 1 \leq k \leq 100 \}\right|\);
[*]The number of inversions as \(\tau(\pi) = \left|\{ (i, j) \mid a_i > a_j, \, 1 \leq i < j \leq 100 \}\right|\);
[*]The century value of the permutation as
\[ f(\pi) = (-1)^{\tau(\pi)} \cdot 2^{\sigma(\pi)} \cdot \frac{\sigma(\pi)}{\sigma(\pi) + 1}. \][/list]
Determine the sum of the century values over all \(100!\) permutations of \(1, 2, \cdots, 100\).
1 reply
EthanWYX2009
Today at 5:27 AM
Photaesthesia
29 minutes ago
J
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Fun primitive root
Mrcuberoot   1
N 42 minutes ago by Hamzaachak
Let $p$ be an odd prime number prove that there exists a positive integer $x$ such that $x$ and $2025x$ are both primitive roots modulo $p$
1 reply
Mrcuberoot
Yesterday at 5:04 AM
Hamzaachak
42 minutes ago
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Sine + Inequality
Oksutok   0
an hour ago
Let $A$ be a set of size $n$ integers. Show that exist some $\theta \in \mathbb{R}$ such that
$$\left|\sum_{a\in A}sin(a\theta)\right |\ge cn^{1/2}(\log n/\log\log n)^{1/2}$$for some constant $c\in\mathbb{R}_{>0}$.
0 replies
Oksutok
an hour ago
0 replies
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INMO 2019 P3
div5252   47
N an hour ago by Golden_Verse
Let $m,n$ be distinct positive integers. Prove that
$$gcd(m,n) + gcd(m+1,n+1) + gcd(m+2,n+2) \le 2|m-n| + 1. $$Further, determine when equality holds.
47 replies
div5252
Jan 20, 2019
Golden_Verse
an hour ago
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Tilted Students Thoroughly Splash Turtle
DottedCaculator   24
N an hour ago by ray66
Source: 2022 USA TSTST #1
Let $n$ be a positive integer. Find the smallest positive integer $k$ such that for any set $S$ of $n$ points in the interior of the unit square, there exists a set of $k$ rectangles such that the following hold:
[list=disc]
[*]The sides of each rectangle are parallel to the sides of the unit square.
[*]Each point in $S$ is not in the interior of any rectangle.
[*]Each point in the interior of the unit square but not in $S$ is in the interior of at least one of the $k$ rectangles
[/list]
(The interior of a polygon does not contain its boundary.)

Holden Mui
24 replies
DottedCaculator
Jun 27, 2022
ray66
an hour ago
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Find all functions
aktyw19   1
N an hour ago by Mathzeus1024
Find all functions $ f: \mathbb R_{+} \to \mathbb R_{+}$ such that for all $ x>0$ and $ 0<y<1$ then $ (1-y)f(x)=f(f(yx)\frac{1-y}{y})$
1 reply
aktyw19
Mar 8, 2014
Mathzeus1024
an hour ago
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Interesting inequality
sqing   3
N an hour ago by sqing
Source: Own
Let $ a,b> 0 ,a^2-ab+b^2=1 . $ Prove that
$$ \frac{a}{a^2+b+1}+\frac{b}{b^2+a+1} \leq \frac{2}{3}$$$$ \frac{a}{b^2+a+1}+\frac{b}{a^2+b+1} < \frac{7}{10}$$Let $ a,b> 0 ,a^2-ab+b^2=\frac{1}{2}. $ Prove that
$$ \frac{a}{b^2+a+1}+\frac{b}{a^2+b+1} \leq \frac{2(3 \sqrt{2}-2)}{7} $$Let $ a,b> 0 ,a^2-ab+b^2=\frac{1}{4}. $ Prove that
$$ \frac{a}{b^2+a+1}+\frac{b}{a^2+b+1} \leq \frac{4}{7}$$Let $ a,b> 0 ,a^2-ab+b^2=\frac{1}{9}. $ Prove that
$$ \frac{a}{b^2+a+1}+\frac{b}{a^2+b+1} \leq \frac{6}{13}$$
3 replies
sqing
Today at 1:56 AM
sqing
an hour ago
J
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Interesting inequality
sqing   6
N an hour ago by sqing
Source: Own
Let $ a,b,c>0,(a+b+1)\left(\frac{1}{a} + \frac{1}{b} +1\right)= 10. $ Prove that
$$ \frac{4\sqrt{2}}{5} \geq\frac{ \sqrt{a} + \sqrt{b} }{a+b+ab}\geq \frac{1 }{2\sqrt{2}}$$$$ \frac{4(1+\sqrt{2})}{5} \geq \frac{ \sqrt{a} + \sqrt{b} +1}{a+b+ab}\geq \frac{1+2\sqrt{2} }{8}$$
6 replies
sqing
Jul 29, 2025
sqing
an hour ago
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Parallelogram
m4thbl3nd3r   6
N an hour ago by Royal_mhyasd
Let $AD$ be the $A-$altitude of the triangle $ABC$ and $T,S$ be foots of perpendicular lines through $D$ on $AB,AC$, respectively. Construct the parallelogram $DTKS$ and altitudes $BE,CF$ of the triangle $ABC$. Prove that $K$ lies on $EF$
6 replies
m4thbl3nd3r
Today at 3:08 AM
Royal_mhyasd
an hour ago
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one variable function
youochange   1
N May 17, 2025 by Fishheadtailbody
$f:\mathbb R-\{0,1\} \to \mathbb R$


$f(x)+f(\frac{1}{1-x})=2x$
1 reply
youochange
May 17, 2025
Fishheadtailbody
May 17, 2025
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one variable function
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Reveal topic tags
function
functional equation
algebra
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youochange
206 posts
youochange
#1 h  May 17, 2025, 2:49 PM • 1 Y
Y by PikaPika999
$f:\mathbb R-\{0,1\} \to \mathbb R$


$f(x)+f(\frac{1}{1-x})=2x$
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The post below has been deleted. Click to close.
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Fishheadtailbody
9 posts
Fishheadtailbody
#2 h  May 17, 2025, 3:13 PM • 1 Y
Y by youochange
Notice that if $g(x) = \frac{1}{1 - x}$, then $g^3(x) = g(x)$.
Hence we can consider the following, which is simply a system of equations with three variables,
\begin{align*} f(x) + f(g(x)) = 2x, \\ f(g(x)) + f(g(g(x)) = 2g(x), \\ f(g^2(x)) + f(x) = 2g^2(x). \end{align*}Thus,
$f(x) = x - g(x) + g^2(x) = x - \frac{1}{1 - x} + \frac{x - 1}{x}$.
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