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k a July Highlights and 2025 AoPS Online Class Information

jwelsh 0

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

Difference of Two Sides

xeroxia 4

Let $M$ and $N$ be points on segment $BC$ of triangle $ABC$ such that $AN$ is an angle bisector and $BN=CM$ . The line perpendicular to $AN$ at $N$ meets the circle with center $A$ and passing through $M$ at $D$ . Show that $DN = |AB - AC|$ .

4 replies

xeroxia

Yesterday at 7:28 AM

Double07

9 minutes ago

Cute Simple Geometry

TigerOnion 0

Source: Cyprus 2024 Regional Competition B Lyceum

Two circles $K_1(O_1, R_1)$ and $K_2(O_2, R_2)$ with $R_1 < R_2$ intersect at points $A$ and $B$ . From point $A$ , we draw a tangent to $K_1$ which intersects $K_2$ at point $E$ . The line $EB$ intersects $K_1$ for the second time at point $K$ (we assume that $K$ lies outside $K_2$ ). From point $K$ , we draw a tangent to $K_2$ which intersects $K_1$ at point $\Gamma$ and touches $K_2$ at point $\Delta$ (with $\Delta$ and $A$ lying on opposite sides of $EK$ ). Let $H$ be the intersection point of $A\Gamma$ with $EK$ , and let $(\omega)$ be the circumcircle of triangle $\triangle AHE$ . If $A\Delta$ intersects $(\omega)$ again at point $N$ , prove that $NH = NE$ .

0 replies

TigerOnion

24 minutes ago

0 replies

Sequence

MF163 4

Source: Baltic Way 2017 Problem 1

Let $a_0,a_1,a_2,...$ be an infinite sequence of real numbers satisfying $\frac{a_{n-1}+a_{n+1}}{2}\geq a_n$ for all positive integers $n$ . Show that $\frac{a_0+a_{n+1}}{2}\geq \frac{a_1+a_2+...+a_n}{n}$ holds for all positive integers $n$ .

4 replies

MF163

Nov 14, 2017

Assassino9931

32 minutes ago

again ineq)

Maksat_B 39

Source: JBMO 2025 pr1

For all positive real numbers $a, b, c$ , prove that
$\frac{(a^2 + bc)^2}{b + c} + \frac{(b^2 + ca)^2}{c + a} + \frac{(c^2 + ab)^2}{a + b} \geq \frac{2abc(a + b + c)^2}{ab + bc + ca}.$
Proposed by Hakan Karakuş, Türkiye

39 replies

Maksat_B

Jun 26, 2025

Ihatecombin

an hour ago

College Math Topics in undergraduate and graduate studies

Topics in undergraduate and graduate studies

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College Math Topics in undergraduate and graduate studies

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k a July Highlights and 2025 AoPS Online Class Information

jwelsh 0

0 replies

jwelsh

Jul 1, 2025

0 replies

Hard sequence problem

P0tat0b0y 0

Please help! This is not homework! But it would be very important to get an answer!
Is there a sequence ${{a}_{n}}$ for which the following 4 conditions are met:
(1) ${{a}_{1}}\le 4-\pi$
(2) $3,(5)-\frac{{{a}_{2}}}{2}\ge \pi$
(3) $\underset{n\to \infty }{\mathop{\lim }}\,\frac{{{a}_{n}}}{n}=0$
(4) ${{(2n+2)}^{2}}{{a}_{n}}\ge {{(2n+1)}^{2}}{{a}_{n+1}}+(n+1)\pi ,\forall n\ge {{n}_{0}}$
Thanks in advance!

0 replies

P0tat0b0y

7 minutes ago

0 replies

?k/nk is an irrational number

fxandi 1

If $\dfrac{n_{k+1}}{n_k} \geq Q > 1$ for $k \in \mathbb{N}$ . Prove that $\displaystyle\sum \dfrac{\varepsilon_k}{n_k}$ is an irrational number.
In the problem above $\{\varepsilon_k\}$ is arbitrary sequence with elements $+1$ and $-1$ .

1 reply

fxandi

6 hours ago

greenturtle3141

4 hours ago

Countable of Z

Tricky123 0

Q) prove that $\mathbb{Z}$ is countable.
Proof:- I have constructed a bijective function like $f:\mathbb{N}\to \mathbb{Z}$ such that
$f(n) = \begin{cases} \frac{n-1}{2} & \text{if n is odd} \\ -\frac{n}{2} & \text{if n is even} \end{cases}$
Now how to visualise this function in this given visualise way
https://youtube.com/shorts/M1dQNwyBjg0?si=cXOM9NR2rv3QIzYq

And extended to Z*Z
How to make guess for the function help please

0 replies

Tricky123

5 hours ago

0 replies

Equality of integral over closed interval

Synchrone 2

Source: Math&Maroc Competition 2025 Day2 Problem 6

Let $f,g : [0,1] \to \mathbb{R}_{>0}$ be two continuous and strictly positive functions. We assume that $\int_0^1 f = \int_0^1 g = 1$ .
Show that there exists a closed interval, $I \subset [0,1]$ such that $\int_I f = \int_I g = \frac{1}{2}$

2 replies

Synchrone

Yesterday at 3:21 PM

Photaesthesia

Today at 2:13 AM

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Problem of the Day 2011 Argentina Team Selection Test

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A wizard kidnaps $31$ members from party $A$ , $28$ members from party $B$ , $23$ members from party $C$ , and $19$ members from party $D$ , keeping them isolated in individual rooms in his castle, where he forces them to work.
Every day, after work, the kidnapped people can walk in the park and talk with each other. However, when three members of three different parties start talking with each other, the wizard reconverts them to the fourth party (there are no conversations with $4$ or more people involved).

a) Find out whether it is possible that, after some time, all of the kidnapped people belong to the same party. If the answer is yes, determine to which party they will belong.
b) Find all quartets of positive integers that add up to $101$ that if they were to be considered the number of members from the four parties, it is possible that, after some time, all of the kidnapped people belong to the same party, under the same rules imposed by the wizard.

Leicich

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Contest of the Week 2025 Moldova Team Selection Test

2025 Moldova Team Selection Test

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Contest of the Week 2025 Moldova Team Selection Test

2025 Moldova Team Selection Test

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2025 Moldova Team Selection Test

Currently under a confidentiality agreement.

Let $A, B, C, D$ be fixed points on this order on a line. Let $\omega$ be a variable circle through $C$ and $D$ and suppose it meets the perpendicular bisector of $CD$ at the points $X$ and $Y$ . Let $Z$ and $T$ be the other points of intersection of $AX$ and $BY$ with $\omega$ . Prove that $ZT$ passes through a fixed point independent of $\omega$ .

Assassino9931

view topic

Currently under a confidentiality agreement.

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k a July Highlights and 2025 AoPS Online Class Information

jwelsh 0

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jwelsh

Jul 1, 2025

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Cannot Load Homework Error

zhenghua 38

Summary of the problem: Some error is preventing me from finishing my homework. My brother is encountering the same problem.
Page URL: Your class homework
Steps to reproduce:
1. Open the classes tab.
2. Open homework
3. A window pops up.
Expected behavior: You should be able to access your homework.
Frequency: 100% after 18:00 PST
Operating system(s): Windows 10
Browser(s), including version: Firefox (and my brother tried google)
Additional information: Screenshot below (cannot upload screenshot for some reason...). As a side note, recently, I've been experiencing extreme lag on AoPS that doesn't seen to occur with other websites.

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zhenghua

Today at 3:14 AM

CertainlyNotaCat

2 hours ago

Emoji ....

princeofmath 7

Does anybody know all the codes for AOPS emojis?

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princeofmath

3 hours ago

Zestra

3 hours ago

My TeXeR Is Not Saving, or Rendering.

AlienGirl05 0

Greetings, Site Support,

I've been working on a TeXeR file. However, all of a sudden, it stopped saving my work, or rendering it. No matter how many times I pressed the save button, I wasn't being notified of my work being saved. Upon reloading the page, it wouldn't render my work (don't worry, I didn't actually lose any of my work; I have it saved in a backup Google Document). The "Render With BBCode" button isn't working either.

Thank you. :)

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AlienGirl05

6 hours ago

0 replies

Failure loading my classroom page

HW8128JFK 1

After going onto the homework page of my introduction to geometry class, I found a strange message popping up, and I don't what in the world it means. It keeps showing up when I refresh it, and I've never seen it before. For a second I thought it stopped because it showed me week 17 homework, but when I switched to week 16, the message popped again, and now it's doing it for week 17 too. I was going to attach a screen shot, but now it's not letting me do that either, (whenever I click on the screen shot I want to upload it shows it's loading for a split second and then disappears, like I never chose a file) so maybe the whole site just really hates me. Anywho, I'm gonna paste in the message below instead, for anyone to try and decipher, because I am fully stumped.

(I'm gonna edit this part out bc there's already another discussion on it and I think it had my IP so better safe than sorry)

1 reply

HW8128JFK

6 hours ago

tintin21

6 hours ago

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All-Russian Olympiad 2025

2025 All-Russian Olympiad Regional Round
All-Russian Olympiad Regional Round 2025

2025 Austrian MO National Competition
Austrian MO National Competition 2025

2025 Austrian MO Regional Competition
Austrian MO Regional Competition 2025

2025 Azerbaijan Junior NMO
For junior (8-9 grades)

2025 Azerbaijan Senior NMO
For seniors (10-11 grades)

2025 Balkan MO
Balkan MO 2025

2025 Bangladesh Mathematical Olympiad
BdMO 2025 National Round Problems

2025 Belarusian National Olympiad
Belarusian National Olympiad 2025

2025 Benelux
2025 Benelux Mathematical Olympiad

2025 Bulgarian Spring Mathematical Competition
Bulgarian Spring Mathematical Competition 2025

2025 Bulgarian Winter Tournament
Bulgarian Winter Tournament 2025

2025 Bundeswettbewerb Mathematik
Bundeswettbewerb Mathematik 2025

2025 Canada Junior National Olympiad
Canada Junior National Olympiad 2025

2025 Canada National Olympiad
Canada National Olympiad 2025

2025 CAPS Match
Czech-Austrian-Polish-Slovak Match 2025; https://oemo.at/OeMO/Downloads/109/

2025 Caucasus Mathematical Olympiad
X Caucasus Mathematical Olympiad

2025 China National Olympiad
2025 China National Olympiad

2025 China Team Selection Test
China Team Selection Test 2025

2025 EGMO
European Girls MO 2025

2025 Euler Olympiad

2025 Francophone Mathematical Olympiad
Math olympiad for the French Speaking, 2025

2025 German National Olympiad
German National Olympiad 2025, Final Round

2025 Greece National Olympiad
2025 Greece National Olympiad

2025 India National Olympiad
India National Olympiad 2025

2025 India STEMS

2025 International Mathematical Olympiad
IMO 2025

2025 International Zhautykov Olympiad
International Zhautykov Olympiad 2025

2025 Iran MO (2nd Round)
National Math Olympiad (Second Round) 2025

2025 Israel National Olympiad (Gillis)
An Olympiad for grades 7-12.

2025 Israel TST

2025 Japan MO Finals
Japan MO Finals 2025

2025 Korea - Final Round
FKMO 2025

2025 Korea Winter Program Practice Test
Korea Winter Program Practice Test 2025

2025 Kyiv City MO Round 1
Problems from first round of Kyiv Mathematical Olympiad 2025

2025 Kyiv City MO Round 2
Problems from second round of Kyiv Mathematical Olympiad 2025

2025 Macedonian Mathematical Olympiad
The problems from the 32nd Macedonian Mathematical Olympiad

2025 Macedonian TST
The problems from the 2025 Macedonian TST

2025 Malaysian APMO Camp Selection Test
Malaysian APMO Camp Selection Test for APMO 2025

2025 Nordic
Nordic 2025

2025 Philippine MO

2025 Poland - First Round

2025 Poland - Second Round
Second Round - Poland 2025

2025 Polish MO Finals
Polish MO Finals 2025

2025 Portugal MO
Portugal MO 2025

2025 Romanian Master of Mathematics
15th RMM 2025

2025 Sharygin Geometry Olympiad
Sharygin Geometry Olympiad 2025

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