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:
All classes start 7:30pm ET/4:30pm PT unless otherwise noted.

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0 replies
jwelsh
Jul 1, 2025
0 replies
k i Suggestion Form
jwelsh   0
May 6, 2021
Hello!

Given the number of suggestions we’ve been receiving, we’re transitioning to a suggestion form. If you have a suggestion for the AoPS website, please submit the Google Form:
Suggestion Form

To keep all new suggestions together, any new suggestion threads posted will be deleted.

Please remember that if you find a bug outside of FTW! (after refreshing to make sure it’s not a glitch), make sure you’re following the How to write a bug report instructions and using the proper format to report the bug.

Please check the FTW! thread for bugs and post any new ones in the For the Win! and Other Games Support Forum.
0 replies
jwelsh
May 6, 2021
0 replies
k i Read me first / How to write a bug report
slester   3
N May 4, 2019 by LauraZed
Greetings, AoPS users!

If you're reading this post, that means you've come across some kind of bug, error, or misbehavior, which nobody likes! To help us developers solve the problem as quickly as possible, we need enough information to understand what happened. Following these guidelines will help us squash those bugs more effectively.

Before submitting a bug report, please confirm the issue exists in other browsers or other computers if you have access to them.

For a list of many common questions and issues, please see our user created FAQ, Community FAQ, or For the Win! FAQ.

What is a bug?
A bug is a misbehavior that is reproducible. If a refresh makes it go away 100% of the time, then it isn't a bug, but rather a glitch. That's when your browser has some strange file cached, or for some reason doesn't render the page like it should. Please don't report glitches, since we generally cannot fix them. A glitch that happens more than a few times, though, could be an intermittent bug.

If something is wrong in the wiki, you can change it! The AoPS Wiki is user-editable, and it may be defaced from time to time. You can revert these changes yourself, but if you notice a particular user defacing the wiki, please let an admin know.

The subject
The subject line should explain as clearly as possible what went wrong.

Bad: Forum doesn't work
Good: Switching between threads quickly shows blank page.

The report
Use this format to report bugs. Be as specific as possible. If you don't know the answer exactly, give us as much information as you know. Attaching a screenshot is helpful if you can take one.

Summary of the problem:
Page URL:
Steps to reproduce:
1.
2.
3.
...
Expected behavior:
Frequency:
Operating system(s):
Browser(s), including version:
Additional information:


If your computer or tablet is school issued, please indicate this under Additional information.

Example
Summary of the problem: When I click back and forth between two threads in the site support section, the content of the threads no longer show up. (See attached screenshot.)
Page URL: http://artofproblemsolving.com/community/c10_site_support
Steps to reproduce:
1. Go to the Site Support forum.
2. Click on any thread.
3. Click quickly on a different thread.
Expected behavior: To see the second thread.
Frequency: Every time
Operating system: Mac OS X
Browser: Chrome and Firefox
Additional information: Only happens in the Site Support forum. My tablet is school issued, but I have the problem at both school and home.

How to take a screenshot
Mac OS X: If you type ⌘+Shift+4, you'll get a "crosshairs" that lets you take a custom screenshot size. Just click and drag to select the area you want to take a picture of. If you type ⌘+Shift+4+space, you can take a screenshot of a specific window. All screenshots will show up on your desktop.

Windows: Hit the Windows logo key+PrtScn, and a screenshot of your entire screen. Alternatively, you can hit Alt+PrtScn to take a screenshot of the currently selected window. All screenshots are saved to the Pictures → Screenshots folder.

Advanced
If you're a bit more comfortable with how browsers work, you can also show us what happens in the JavaScript console.

In Chrome, type CTRL+Shift+J (Windows, Linux) or ⌘+Option+J (Mac).
In Firefox, type CTRL+Shift+K (Windows, Linux) or ⌘+Option+K (Mac).
In Internet Explorer, it's the F12 key.
In Safari, first enable the Develop menu: Preferences → Advanced, click "Show Develop menu in menu bar." Then either go to Develop → Show Error console or type Option+⌘+C.

It'll look something like this:
IMAGE
3 replies
slester
Apr 9, 2015
LauraZed
May 4, 2019
k i Community Safety
dcouchman   0
Jan 18, 2018
If you find content on the AoPS Community that makes you concerned for a user's health or safety, please alert AoPS Administrators using the report button (Z) or by emailing sheriff@aops.com . You should provide a description of the content and a link in your message. If it's an emergency, call 911 or whatever the local emergency services are in your country.

Please also use those steps to alert us if bullying behavior is being directed at you or another user. Content that is "unlawful, harmful, threatening, abusive, harassing, tortuous, defamatory, vulgar, obscene, libelous, invasive of another's privacy, hateful, or racially, ethnically or otherwise objectionable" (AoPS Terms of Service 5.d) or that otherwise bullies people is not tolerated on AoPS, and accounts that post such content may be terminated or suspended.
0 replies
dcouchman
Jan 18, 2018
0 replies
AOPS MO Introduce
MathMaxGreat   45
N 4 minutes ago by lendsarctix280
$AOPS MO$

Problems: post it as a private message to me or @jerryZYang, please post it in $LATEX$ and have answers

6 Problems for two rounds, easier than $IMO$

If you want to do the problems or be interested, reply ’+1’
Want to post a problem reply’+2’ and message me
Want to be in the problem selection committee, reply’+3’
45 replies
MathMaxGreat
Today at 1:04 AM
lendsarctix280
4 minutes ago
Problem 4
codyj   94
N 40 minutes ago by OronSH
Source: IMO 2015 #4
Triangle $ABC$ has circumcircle $\Omega$ and circumcenter $O$. A circle $\Gamma$ with center $A$ intersects the segment $BC$ at points $D$ and $E$, such that $B$, $D$, $E$, and $C$ are all different and lie on line $BC$ in this order. Let $F$ and $G$ be the points of intersection of $\Gamma$ and $\Omega$, such that $A$, $F$, $B$, $C$, and $G$ lie on $\Omega$ in this order. Let $K$ be the second point of intersection of the circumcircle of triangle $BDF$ and the segment $AB$. Let $L$ be the second point of intersection of the circumcircle of triangle $CGE$ and the segment $CA$.

Suppose that the lines $FK$ and $GL$ are different and intersect at the point $X$. Prove that $X$ lies on the line $AO$.

Proposed by Greece
94 replies
codyj
Jul 11, 2015
OronSH
40 minutes ago
Olympiad Algebra Final Boss
Iveela   5
N 43 minutes ago by Amirreza.J
Source: 2025 IRN-MNG Friendly Competition
Determine the smallest constant $C$ such that for all sequences $x_1, x_2, \dots, x_n$ of positive real numbers
\[\sum\limits_{1 \leq i, j \leq n} \left\{\frac{x_i}{x_j} \right\} \leq Cn^2.\]As usual, $\{x\}$ denotes the fractional part of $x$.
5 replies
Iveela
Jun 8, 2025
Amirreza.J
43 minutes ago
2^x + 3^y a perfect square, find positive integers x,y
parmenides51   13
N an hour ago by SirAppel
Source: JBMO Shortlist 2017 NT3
Find all pairs of positive integers $(x,y)$ such that $2^x + 3^y$ is a perfect square.
13 replies
parmenides51
Jul 25, 2018
SirAppel
an hour ago
k Cannot post PHP
char0221   7
N Jul 8, 2025 by huolongguo10
Summary of the problem: If I try to post anything with PHP (a coding language), it
Page URL: In any forum or private messages
Steps to reproduce:
1. Create a post.
2. Put some PHP inside, can't give example
Expected behavior: Should post the message
Frequency: 100%
Operating system(s): macOS Sequoia 15.2.1
Browser(s), including version: Safari
Additional information: See attachments
7 replies
char0221
Apr 30, 2025
huolongguo10
Jul 8, 2025
k AoPS Wiki being really slow
SwordAxe   2
N Jul 8, 2025 by SwordAxe
Summary of the problem: AoPS wiki is being really slow, when I'm loading up amc 10 questions it takes around 30 seconds to load, but other parts of the site aren't slow
Page URL: aopswiki
Steps to reproduce:
1. go onto aops wiki and searh an amc 10 question or any type of question or page

Expected behavior: being slow
Frequency: almost always, sometimes it loads fast
Operating system(s): Windows 10
Browser(s), including version: newest microsoft edge
Additional information:
2 replies
SwordAxe
Jul 8, 2025
SwordAxe
Jul 8, 2025
PAGE NOT SCROLLING DOWN ALL THE WAY; GETS STUCK 1.5 POSTS BEFORE THREAD END
CurlyFalcon55   31
N Jul 8, 2025 by PikaPika999
Summary of the problem:
Whenever I push the big, top right "Reply" button– NOT THE SMALL LITTLE "QUICK REPLY" BAR ON THE BOTTOM, and reply, I can't scroll down all the way to the bottom of the thread that I just posted in. Sometimes, very rarely though, I doesn't scroll down all the way when I push the "Quick Reply" bar.


-----Page URL:
Anywhere in the "Introduction to Number Theory" forum in AoPS


-----Steps to reproduce:

1. Go to the link above.

2. Click any thread or create a new one.

3. Click the "Reply" button, not the quick reply bar.

4. Type a reply.

5. Push the "Submit" button.

6. Try to scroll down. if you’re lucky, the problem will reproduce.

-----Expected behavior:
To scroll all the way down.


-----Frequency:
Almost every time I push the big "Reply" button, sometimes (but rarely) when I push the "Quick Reply" bar.


-----Operating system(s):
macOS Sequoia 15.4.1


-----Browser(s), including version:
Chrome, (I don't know what version)


-----Additional information:
-It rarely does it in the "Quick Reply" bar.
-It almost always does it in the big "Reply" button on the top right.


-----Can anyone reproduce?
31 replies
CurlyFalcon55
Jun 1, 2025
PikaPika999
Jul 8, 2025
k I can't type into boxes during lessons
HADP   7
N Jul 7, 2025 by jlacosta
Recently, I have been experiencing an issue where i can't type into the boxes that professors post during a lesson. I can't input an answer and i've tried refreshing multiple times and it doesn't work. This has happened in many lessons in a row. Please help! Also, is there a way to make Aops dark mode?
7 replies
HADP
Jul 1, 2025
jlacosta
Jul 7, 2025
k Happy 4th of July!!!
pingpongmerrily   207
N Jul 7, 2025 by pingpongmerrily
AoPS is taking classes off for it, so I think we should celebrate it...
[rule]
[center]IMAGE
[rule]
:trampoline: Happy 4th of July!!!
207 replies
pingpongmerrily
Jul 4, 2025
pingpongmerrily
Jul 7, 2025
k Just so SS knows....
jmr2010   10
N Jul 7, 2025 by CuriousMathBoy72
If you open a PM, specify yourself, and put something there, then hit send, it says you are automatically included and if you want to talk to yourself, leave the to box blank. If you leave the box blank, it won't send the PM because the to box is blank
10 replies
jmr2010
Jul 2, 2025
CuriousMathBoy72
Jul 7, 2025
k I've been wrongfully disabled
K124659   5
N Jul 5, 2025 by K124659
Summary of the problem: My alcumus account has been "banned"
Page URL: Anywhere on alcumus
Steps to reproduce (I reccomend you don't):
1. Answer questions extremely quickly whilst spamming the enter button
Expected behavior: You'll get banned. I believe this is some sort of system to disable bots.
Frequency: I'm not sure.
Operating system(s): Acer 311 Chromebook
Browser(s), including version: Google Chrome
Additional information: I believe this is due to the fact that I knew most of the questions either by heart or because they are extremely easy (It was multiplying fractions in pre-algebra that got me banned), and as a result I was going too fast and was spamming inputs into the system, similar to a bot. I wasn't using a bot, however. I'm not even grinding HOF- I had broken record as a quest with my problem streak being 399 previously, so my usage of the same subject to get a bunch correct was lowk necessary ngl.
5 replies
K124659
Jul 5, 2025
K124659
Jul 5, 2025
k Searching for keywords
Cats_on_a_computer   1
N Jul 4, 2025 by Demetri
Quick question because I’m new here, I keep searching keywords in the community search functions to check if they’ve been posted before, and every time it ended up not finding the original, leading to a repost. This has happened so many times I’ve decided not to post anymore until I can somehow find a way to efficiently search for it.
I accidentally reposted a Putnam 2006, even though I searched for the keywords in the question, but since I didn’t know it was a Putnam problem (I had seen it somewhere else), I didn’t manage to find the problem so unknowingly reposted it.
1 reply
Cats_on_a_computer
Jul 4, 2025
Demetri
Jul 4, 2025
k REQUEST LOCK
KangarooPrecise   25
N Jul 3, 2025 by ahxun2006
On my username, why is there 3 dots after it, when 2 letters are missing?
25 replies
KangarooPrecise
Jul 2, 2025
ahxun2006
Jul 3, 2025
k FOUR PROBLEMS
CurlyFalcon55   4
N Jul 3, 2025 by CurlyFalcon55
There are four homework problems that are REALLY BUGGING ME because I CANNOT FIND THE ANSWER. :mad:
Can I make a forum where I can ask questions about these problems? Can I make one as long as it follows the AoPS Honor Code and the Terms of Service? (I can explicitly state that in the forum.)
If I have to I can make it private with a few select people. Or can I make a blog version? ;)

I know this was kind of already adressed in the "Is there a forum?" thread, but that was for existing forums. This is for a potential new public/private forum. :)

Read before replying

IMAGE
4 replies
CurlyFalcon55
Jul 2, 2025
CurlyFalcon55
Jul 3, 2025
Decimal functions in binary
Pranav1056   3
N May 23, 2025 by ihategeo_1969
Source: India TST 2023 Day 3 P1
Let $\mathbb{N}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $f(x) + y$ and $f(y) + x$ have the same number of $1$'s in their binary representations, for any $x,y \in \mathbb{N}$.
3 replies
Pranav1056
Jul 9, 2023
ihategeo_1969
May 23, 2025
Decimal functions in binary
G H J
G H BBookmark kLocked kLocked NReply
Source: India TST 2023 Day 3 P1
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Pranav1056
35 posts
#1 • 4 Y
Y by GeoKing, CahitArf, Supercali, Siddharth03
Let $\mathbb{N}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $f(x) + y$ and $f(y) + x$ have the same number of $1$'s in their binary representations, for any $x,y \in \mathbb{N}$.
This post has been edited 2 times. Last edited by Pranav1056, Jul 9, 2023, 6:22 AM
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Supercali
1263 posts
#2 • 2 Y
Y by Om245, thepassionatepotato
A story about this problem: It was originally meant to be D3 P3 (basically the hardest problem in the TSTs), but a few days before the test, some of us found an easier solution while trying. Hence the problem had to be demoted to D3 P1. D4 P3 at that time, which was a very hard geo, was shifted to D3 P3 (I think it was more suitable for that position anyway), and we had to use the shortlist for D4 P3. Anyway, I think this is a very cute problem.

Here is the solution that we found:

For $n\in\mathbb N$, let $d(n)$ denote the number of $1$'s in the binary representation of $n$. Let $P(x,y)$ denote the statement that $f(x)+y$ and $f(y)+x$ have the same number of $1$'s in their binary representation.

Claim 1: For any $y,n \in \mathbb{N}$ with $2^n>f(y)$, $f(2^n-f(y))+y$ is a power of two.
Proof: $P(2^n-f(y),y)$ gives us that $2^n$ and $f(2^n-f(y))+y$ have the same number of $1$'s, and the former has exactly one $1$, so $f(2^n-f(y))+y$ has exactly one $1$, from which the claim follows. $\blacksquare$

Claim 2: $f(y+2^k)-f(y)$ is a power of two for any $k \geq 0$ and $y \geq 2^k$.
Proof: Choose an $n$ such that $n>1000+\log_2(10+|f(y+2^k)-f(y)|)$. By Claim 1, $f(2^n-f(y))=2^t-y >0$ for some $t$ with $t \geq k+1$. Therefore $P(2^n-f(y),y+2^k)$ gives $$d(2^n-f(y)+f(y+2^k))=d(2^t+2^k)=2$$since $t \geq k+1$. If $f(y)=f(y+2^k)$, then LHS is $d(2^n)=1$, contradiction! If $f(y)>f(y+2^k)$, and $f(y)-f(y+2^k)$ has $m<\log_2(10+|f(y+2^k)-f(y)|)$ digits, then $$d(2^n-f(y)+f(y+2^k)) \geq n-m-1 \geq 999>2$$since $2^n-f(y)+f(y+2^k)$ starts with at least $n-m-1$ ones, contradiction! Therefore $f(y+2^k)>f(y)$, and since $n$ is bigger that the number of digits in $f(y+2^k)-f(y)$, there is no carry-over, so
$$2=d(2^n-f(y)+f(y+2^k))=1+d(f(y+2^k)-f(y))$$which gives us $f(y+2^k)-f(y)$ is a power of $2$, as required. $\blacksquare$


Claim 2 gives us $f(y+1)-f(y)=2^{t(y)}$ for some $t(y)$, for all $y$. But for $y \geq 2$, $f(y+2)-f(y)$ is also a power of two $\implies$ $2^{t(y)}+2^{t(y+1)}$ is a power of two, which is only possible if $t(y)=t(y+1)$ for all $y \geq 2$. Therefore $f(y+1)-f(y)$ is a constant power of two for all $y \geq 2$, say $2^k$. This gives us $f(y)=2^ky+c$ for some constant $c$, for all $y \geq 2$. Putting this in Claim 1, we get
$$2^{k+n}-(2^{2k}-1)y-(2^k-1)c$$is a power of two for any $y \geq 2$ and any sufficiently large $n$. This is only possible if, for all $y \geq 2$,
$$(2^{2k}-1)y+(2^k-1)c=0$$$$\iff (2^k-1)((2^k+1)y+c)=0$$which can only hold for all $y \geq 2$ if $2^k=1$, i.e., $f(y)=y+c$ for all $y \geq 2$. But Claim 1 for $y=1$ and large $n$ gives
$$2^n-f(1)+1+c$$is a power of two for all sufficiently large $n$, which is only possible if $f(1)=1+c$. Therefore the only solutions are
$$\boxed{f(x)=x+c \ \ \forall x \in \mathbb{N}}$$where $c$ is a non-negative integer.
This post has been edited 3 times. Last edited by Supercali, Jul 27, 2023, 7:50 AM
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i3435
1351 posts
#3 • 2 Y
Y by GeoKing, Om245
$P(2^n-f(x),x)$ means $x=2^k-f(2^n-f(x))$ for some $k$, for all $x,n$. $P(2^a+2^k-f(x),x)$ means that $f(2^a+2^k-f(x))+x$ is of the form $2^c+2^d$, where $c\neq d$ when $a\neq k$ and $c=d$ when $a=k$. Replacing $x$ in the previous equation with $2^n-f(x)$, $f(2^a+x)+2^n-f(x)$ either has one or two ones in its binary representation. If you make $n$ large, we get that $f(2^a+x)-f(x)$ is a power of two for all $a,x$. In the same manner as the previous post, you can get $f(x)=2^kx+c$ for some $k,c$. $P(x,2^kx)$ means $2^{k+1}x+c$ and $(2^{2k}+1)x+c$ have the same number of $1$'s in their binary representation. If $x$ is a large power of $2$, then the second one will have one more $1$ than the first one unless $k=0$. Thus $k=0$ and $f(x)=x+c$, which works.
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ihategeo_1969
274 posts
#4
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Let $P(x,y)$ denote the assertion. Call $2$ such numbers quirky.

$P(2^n-f(y),y)$ gives us $f(2^n-f(y))+y=2^{g(n,y)}$ for any $n>f(y)$ where $g: \mathbb{N}^2 \to \mathbb N$ is a function. See that $g(n,y)$ is unbounded.

Claim: $f(x+2^\ell)-f(x)$ is a power of $2$ for any $\ell \ge 0$ and $x \ge 2^\ell$.
Proof: $P(x+2^\ell,2^n-f(x))$ gives us that $f(x+2^\ell)+2^n-f(x)$ and $2^{g(n,x)}+2^\ell$ are quirky.

Now $2^{g(n,x)}>2^\ell$ so $f(x+2^\ell)+2^n-f(x)=2^{g_1(n,x,\ell)}+2^{g_2(n,x,\ell)}$ where $g_1$, $g_2: \mathbb{N}^3 \to \mathbb Z _{\ge 0}$ and $g_1(n,x,\ell) \neq g_2(n,x,\ell)$. Fix $x$ and $\ell$ and we will abuse some notation by letting $g_i(n,x,\ell)=g_i(n)$ because I am lazy. So we have \begin{align*}
& 2^{g_1(n)}+2^{g_2(n)}-2^n \text{ is constant} \\
\implies & 2^{g_1(n)}+2^{g_2(n)}+2^m=2^{g_1(m)}+2^{g_2(m)}+2^n
\end{align*}Say $m>ng_1(n)g_2(n)$ and $g_1(n)$, $g_2(n)$, $m$ are all distinct and so is $g_1(m)$, $g_2(m)$. If $n=g_2(m)$ then LHS have $3$ $1$'s in their binary representation but RHS has atmost $2$.

Now as $n \neq m$ so $n \in \{g_1(n),g_2(n)\}$ and hence we get that $2^{g_1(n)}+2^{g_2(n)}-2^n$ is a power of $2$ and so we are done. $\square$

Choose $\ell=0$ and $1$ and easily get that $f(x+1)-f(x)$ is a constant power of $2$ for $x \ge 2$. By a bit case bash we get that that must be $1$. And similarly we get that $f(2)-f(1)=1$ as well.

Hence the only solution is $\boxed{f(x) \equiv x+c \text{ } \forall \text{ }x \in \mathbb{N}}$ where $c \ge 0$; which obviously works.
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