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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've been wrongfully banned
WalterMitchell   4
N 4 hours ago by jlacosta
I was randomly doing Prealg, and this message popped up (see below). I wasn't using a bot or anything. Did I do something wrong? or is it just a glitch?
4 replies
WalterMitchell
Friday at 9:16 PM
jlacosta
4 hours ago
define $f_n(x)$ recursively by
Martin.s   1
N Today at 12:34 PM by Martin.s
Let $f_1(x) = 2\pi \sin (x)$. For $n > 1$, define $f_n(x)$ recursively by
\[
  f_n(x) = 2\pi \sin(f_{n-1}(x)).
\]How many intervals $[a, b]$ are there such that
$\quad \bullet \ $ $0 \le a < b \le 2\pi$,
$\quad \bullet \ $ $f_6(a) = -2\pi$,
$\quad \bullet \ $ $f_6(b)=2\pi$,
$\quad \bullet \ $ and $f_6$ is increasing on $[a, b]$?
1 reply
Martin.s
Aug 25, 2024
Martin.s
Today at 12:34 PM
Bijection and Series
Natrium   1
N Today at 11:47 AM by alexheinis
Source: Inspired by ICMC 2008
Let $f:\mathbb{N}\to\mathbb{N}$ be a bijection. Prove or disprove the following claims:
[list]
[*] $\sum_{n=1}^\infty\frac{n}{f(n)(n+f(n))}=\infty,$ for each such $f,$ [/*]
[*] $\sum_{n=1}^\infty\frac{f(n)}{n(n+f(n))}=\infty,$ for each such $f.$[/*]
[/list]
1 reply
Natrium
Today at 9:20 AM
alexheinis
Today at 11:47 AM
Congruence solvable for all primes $p$
chandru1   0
Today at 11:42 AM
Show that for all primes except the prime $p=3$, the equation $x^2+3y^2 \equiv -4 \pmod{p}$ is always solvable. One can see that for the prime $p=2$, $(x,y)=(0,0)$ are the solutions. For $p=3$ case we see that the equation becomes $x^2 \equiv 2 \pmod{3}$ and this doesn't have any solutions. The case $p>3$ can be handled in $2$ cases. The first case where $p \equiv 1 \pmod{4}$ and $p \equiv 3 \pmod{4}$. From the well known facts that $-1$ and $4$ are quadratic residues for primes of the form $1 \pmod{4}$ we see that the congruence $x^2 + 3y^2 \equiv -4 \pmod{p}$ is solvable (of course with $y=0$). What to do for the case $3 \pmod{4}$? Should we consider the sets $\{x^2 (\pmod p)\}$ and $\{-4-3y^{2} (\pmod p)\}$ and show that they intersect? I haven't tried this line of thought though, but it would nice to have an argument which works specifically for the case $p\equiv 3\pmod{4}$.
0 replies
chandru1
Today at 11:42 AM
0 replies
Vinogradov Thm
EthanWYX2009   0
Today at 9:56 AM
Source: 2024 September 谜之竞赛-6
Show that for any positive real number \(\varepsilon\), there exists a positive integer \(K\) such that the following holds:

For any positive integer \(k \geq K\), there exist a positive integer \(s \leq (1 + \varepsilon) k^2 \ln k\) and \(2s\) positive integers $x_1,$ $x_2,$ $\cdots,$ $x_s,$ $y_1,$ $y_2,$ $\cdots,$ $y_s$ such that for all integers \(1 \leq j \leq k\),
\[\sum_{i=1}^s x_i^j = \sum_{i=1}^s y_i^j,\]but
\[\sum_{i=1}^s x_i^{k+1} \neq \sum_{i=1}^s y_i^{k+1}.\]Proposed by Mucong Sun, Tsinghua University
0 replies
1 viewing
EthanWYX2009
Today at 9:56 AM
0 replies
Polynomial FLT
EthanWYX2009   0
Today at 8:42 AM
Source: 2025 New Year 谜之竞赛-3
Find all rational numbers \( r \) such that there exist complex-coefficient polynomials \( P, Q \) satisfying
\[\{z \in \mathbb{C}: |\operatorname{Re}(z)|^r + |\operatorname{Im}(z)|^r = 1\}[= \left\{ \frac{P(w)}{Q(w)} : w \in \mathbb{C}, |w| = 1 \text{ and } Q(w) \neq 0 \right\}.\]Created by Cheng Jiang, Massachusetts Institute of Technology
0 replies
EthanWYX2009
Today at 8:42 AM
0 replies
Compute the limit
Darealzolt   2
N Today at 7:33 AM by kurumi3rd
Compute the value of the limit
\[
\lim_{n \rightarrow \infty} \frac{1}{n} \left( 1 + \frac{2}{1 + \sqrt{2}} + \frac{3}{1 + \sqrt{2} + \sqrt{3}} + \dots + \frac{n}{1 + \sqrt{2} + \sqrt{3} + \dots + \sqrt{n}} \right)
\]
2 replies
Darealzolt
Today at 1:55 AM
kurumi3rd
Today at 7:33 AM
Wrong Last Poster
ibmo0907   10
N Today at 4:39 AM by ibmo0907
Summary of the problem: The last person to post is wrong in the sidebar.
Page URL: link
Steps to reproduce:
1. Go to the thread
2. Scroll to the bottom
3. The last person to post is $LaTeX$, but in the sidebar, it says that the last poster is phiReKaLk6781.
...
Expected behavior: It should show $LaTeX$ as the last poster, in the sidebar.
Frequency: Always

10 replies
ibmo0907
Jul 18, 2025
ibmo0907
Today at 4:39 AM
k I can edit posts I shouldn't be able to
WalterMitchell   1
N Today at 3:06 AM by PuppyPenguinDolphin
So, I was on this alcumus thread about hof.. and i realized that the edit icon showed up on aidan's post, even though i'm not aidan0626 (see below).

i have no idea why this happened and it was only for aidan0626...

edit: it has disappeared, but i feel that i should keep this thread up because its a pretty serious glitch and could be abused. heres what i think happened: i deleted my post, then the server lagged, so it thought that my post hadn't gotten deleted, and that i should be able to edit the last post
1 reply
WalterMitchell
Today at 2:40 AM
PuppyPenguinDolphin
Today at 3:06 AM
Compute the integral
Darealzolt   0
Today at 2:11 AM
Determine the value of
\[
\int_{0}^{2022\pi}\sqrt{\frac{\cos{2x}-\cos{2\pi}(2022)}{\cos{2x}+1}}dx
\]
0 replies
Darealzolt
Today at 2:11 AM
0 replies
Spectral radius
ILOVEMYFAMILY   2
N Today at 12:16 AM by GreenKeeper
Let $A \in \mathbb{R}^{n \times n}$. The spectral radius of $A$, denoted by $\rho(A)$, is defined as
\[
\rho(A) = \max_i |\lambda_i|
\]where $\lambda_i$ are all the eigenvalues of the matrix $A$.
Let $A \in \mathbb{R}^{n \times n}$. There exists a norm $\|\cdot\|$ such that $\|A\| < 1$ if and only if the spectral radius of $A$ satisfies the condition $\rho(A) < 1$.
2 replies
ILOVEMYFAMILY
Jul 18, 2025
GreenKeeper
Today at 12:16 AM
Derivative problem with nonnegative domain
EmilXM   4
N Yesterday at 9:22 PM by EmilXM
Source: Mock AYT (Turkish entrance exam)
Let $f:\mathbb{R}^+\cup\{0\}\rightarrow\mathbb{R}$ be a differentiable function. If $f(0)=3$, $f'(0)=0$ and $(f(x)-1)f''(x)=x+5$ for all $x\geq0$. Which of the followings are necessarily true:
$i) f'(2)\leq 6$
$ii) f(2)\leq\frac{26}{3}$
$iii)$ f is strictly increasing
4 replies
EmilXM
Yesterday at 6:43 PM
EmilXM
Yesterday at 9:22 PM
Lagrange interpolation
ILOVEMYFAMILY   2
N Yesterday at 4:23 PM by paxtonw
Given a domain $\Omega \subset \mathbb{R}^2$ defined by a closed curve as shown below:
Present a method to approximate the area of the region bounded by the given curve. Clearly describe each computational step and the mathematical theory used at each step.
2 replies
ILOVEMYFAMILY
Yesterday at 11:06 AM
paxtonw
Yesterday at 4:23 PM
k Just a screenshot qustion
DramaticPython94   2
N Jul 18, 2025 by CuriousMathBoy72
How do you add a screenshot to a post? I am on a windows computer :dry:
2 replies
DramaticPython94
Jul 18, 2025
CuriousMathBoy72
Jul 18, 2025
k Reaper bug
AmberTiger79   6
N Apr 1, 2025 by AmberTiger79
So, i went to the reaper page, and it is saying that i was the least person to reap, but i never reaped. Is this supposed to happen?
6 replies
AmberTiger79
Apr 1, 2025
AmberTiger79
Apr 1, 2025
Reaper bug
G H J
G H BBookmark kLocked kLocked NReply
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
AmberTiger79
229 posts
#1
Y by
So, i went to the reaper page, and it is saying that i was the least person to reap, but i never reaped. Is this supposed to happen?
Z Y
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AmberTiger79
229 posts
#2
Y by
And even after someone has reaped, it still won't allow me to reap
Attachments:
This post has been edited 1 time. Last edited by AmberTiger79, Apr 1, 2025, 1:45 PM
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k1glaucus
694 posts
#3
Y by
maybe hard refresh or restart? i cant reproduce
Z Y
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AmberTiger79
229 posts
#4
Y by
still doesn't work
Z Y
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plum28
162 posts
#6
Y by
Maybe it's an April fools especially for you

jk maybe try forcequitting if you're on an ipad or xing out the tab or sign out sign in.
Z Y
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k1glaucus
694 posts
#7
Y by
it shows the screenshot you attached whne the page is loading but not fully loaded (i can see it for a second when I refresh), but that doesnt seewm to be your problem
This post has been edited 1 time. Last edited by k1glaucus, Apr 1, 2025, 4:50 PM
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AmberTiger79
229 posts
#8 • 1 Y
Y by Major_Monogram
Major_Monogram wrote:
H.Wires31 wrote:
Are you at Rocky Point academy? their electronouniforms screw up internet sometimes

no, i am not
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