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k a June Highlights and 2025 AoPS Online Class Information
jlacosta   0
Jun 2, 2025
Congratulations to all the mathletes who competed at National MATHCOUNTS! If you missed the exciting Countdown Round, you can watch the video at this link. Are you interested in training for MATHCOUNTS or AMC 10 contests? How would you like to train for these math competitions in half the time? We have accelerated sections which meet twice per week instead of once starting on July 8th (7:30pm ET). These sections fill quickly so enroll today!

[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC 10 Problem Series[/list]
For those interested in Olympiad level training in math, computer science, physics, and chemistry, be sure to enroll in our WOOT courses before August 19th to take advantage of early bird pricing!

Summer camps are starting this month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have a transformative summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
[list][*]June 5th, Thursday, 7:30pm ET: Open Discussion with Ben Kornell and Andrew Sutherland, Art of Problem Solving's incoming CEO Ben Kornell and CPO Andrew Sutherland host an Ask Me Anything-style chat. Come ask your questions and get to know our incoming CEO & CPO!
[*]June 9th, Monday, 7:30pm ET, Game Jam: Operation Shuffle!, Come join us to play our second round of Operation Shuffle! If you enjoy number sense, logic, and a healthy dose of luck, this is the game for you. No specific math background is required; all are welcome.[/list]
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0 replies
jlacosta
Jun 2, 2025
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
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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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Problem 5
blug   4
N an hour ago by Sir_Cumcircle
Source: Czech-Polish-Slovak Junior Match 2025 Problem 5
For every integer $n\geq 1$ prove that
$$\frac{1}{n+1}-\frac{2}{n+2}+\frac{3}{n+3}-\frac{4}{n+4}+...+\frac{2n-1}{3n-1}>\frac{1}{3}.$$
4 replies
blug
May 19, 2025
Sir_Cumcircle
an hour ago
J
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Cool integer FE
Rijul saini   2
N 2 hours ago by ZVFrozel
Source: LMAO Revenge 2025 Day 1 Problem 1
Alice has a function $f : \mathbb N \rightarrow \mathbb N$ such that for all naturals $a, b$ the function satisfies:
\[a + b \mid a^{f(a)} + b^{f(b)} \]Bob wants to find all possible functions Alice could have. Help Bob and find all functions that Alice could have.
2 replies
Rijul saini
Yesterday at 7:06 PM
ZVFrozel
2 hours ago
J
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star on a quilt
derekwang2048   22
N 2 hours ago by RedFireTruck
Source: 2025 AMC 8 #1
The eight-pointed star, shown in the figure below, is a popular quilting pattern. What percent of the entire $4$-by-$4$ grid is covered by the star?

$\textbf{(A)}\ 40\qquad \textbf{(B)}\ 50\qquad \textbf{(C)}\ 60\qquad \textbf{(D)}\ 75\qquad \textbf{(E)}\ 80$
IMAGE

Thank you @zhenghua for the diagram!
22 replies
derekwang2048
Jan 30, 2025
RedFireTruck
2 hours ago
J
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Frustration with Olympiad Geo
gulab_jamun   18
N 2 hours ago by cooljoseph
Ok, so right now, I am doing the EGMO book by Evan Chen, but when it comes to problems, there are some that just genuinely frustrate me and I don't know how to deal with them. For example, I've spent 1.5 hrs on the second to last question in chapter 2, and used all the hints, and I still am stuck. It just frustrates me incredibly. Any tips on managing this? (or.... am I js crashing out too much?)
18 replies
gulab_jamun
May 29, 2025
cooljoseph
2 hours ago
J
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A beautiful collinearity regarding three wonderful points
math_pi_rate   10
N 2 hours ago by alexanderchew
Source: Own
Let $\triangle DEF$ be the medial triangle of an acute-angle triangle $\triangle ABC$. Suppose the line through $A$ perpendicular to $AB$ meet $EF$ at $A_B$. Define $A_C,B_A,B_C,C_A,C_B$ analogously. Let $B_CC_B \cap BC=X_A$. Similarly define $X_B$ and $X_C$. Suppose the circle with diameter $BC$ meet the $A$-altitude at $A'$, where $A'$ lies inside $\triangle ABC$. Define $B'$ and $C'$ similarly. Let $N$ be the circumcenter of $\triangle DEF$, and let $\omega_A$ be the circle with diameter $X_AN$, which meets $\odot (X_A,A')$ at $A_1,A_2$. Similarly define $\omega_B,B_1,B_2$ and $\omega_C,C_1,C_2$.
1) Show that $X_A,X_B,X_C$ are collinear.
2) Prove that $A_1,A_2,B_1,B_2,C_1,C_2$ lie on a circle centered at $N$.
3) Prove that $\omega_A,\omega_B,\omega_C$ are coaxial.
4) Show that the line joining $X_A,X_B,X_C$ is perpendicular to the radical axis of $\omega_A,\omega_B,\omega_C$.
10 replies
math_pi_rate
Nov 8, 2018
alexanderchew
2 hours ago
J
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Tricky FE
Rijul saini   4
N 2 hours ago by YaoAOPS
Source: LMAO 2025 Day 1 Problem 1
Let $\mathbb{R}$ denote the set of all real numbers. Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that
$$f(xy) + f(f(y)) = f((x + 1)f(y))$$for all real numbers $x$, $y$.

Proposed by MV Adhitya and Kanav Talwar
4 replies
Rijul saini
Yesterday at 6:58 PM
YaoAOPS
2 hours ago
J
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Quotient of Polynomials is Quadratic
tastymath75025   26
N 2 hours ago by pi271828
Source: USA TSTST 2017 Problem 3, by Linus Hamilton and Calvin Deng
Consider solutions to the equation \[x^2-cx+1 = \dfrac{f(x)}{g(x)},\]where $f$ and $g$ are polynomials with nonnegative real coefficients. For each $c>0$, determine the minimum possible degree of $f$, or show that no such $f,g$ exist.

Proposed by Linus Hamilton and Calvin Deng
26 replies
tastymath75025
Jun 29, 2017
pi271828
2 hours ago
J
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MOP Emails Out! (2025)
Mathandski   121
N 2 hours ago by Schintalpati
What an emotional roller coaster the past 34 days have been.

Congrats to all that qualified!
121 replies
Mathandski
Apr 22, 2025
Schintalpati
2 hours ago
J
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Bugs Bunny at it again
Rijul saini   4
N 2 hours ago by ThatApollo777
Source: LMAO 2025 Day 2 Problem 1
Bugs Bunny wants to choose a number $k$ such that every collection of $k$ consecutive positive integers contains an integer whose sum of digits is divisible by $2025$.

Find the smallest positive integer $k$ for which he can do this, or prove that none exist.

Proposed by Saikat Debnath and MV Adhitya
4 replies
Rijul saini
Yesterday at 7:01 PM
ThatApollo777
2 hours ago
J
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Orthocenters equidistant from circumcenter
Rijul saini   5
N 2 hours ago by YaoAOPS
Source: India IMOTC 2025 Day 1 Problem 2
In triangle $ABC$, consider points $A_1,A_2$ on line $BC$ such that $A_1,B,C,A_2$ are in that order and $A_1B=AC$ and $CA_2=AB$. Similarly consider points $B_1,B_2$ on line $AC$, and $C_1,C_2$ on line $AB$. Prove that orthocenters of triangles $A_1B_1C_1$ and $A_2B_2C_2$ are equidistant from the circumcenter of $ABC$.

Proposed by Shantanu Nene
5 replies
Rijul saini
Yesterday at 6:31 PM
YaoAOPS
2 hours ago
J
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Six variables (2)
Nguyenhuyen_AG   1
N 2 hours ago by lbh_qys
Let $a, \, b, \,c, \, x, \, y, \, z$ be six positive real numbers. Prove that
\[a^2+b^2+c^2+\frac{4(ax+by+cz)\sqrt{ab+bc+ca}}{x+y+z} \geqslant 2(ab+bc+ca).\]
1 reply
Nguyenhuyen_AG
3 hours ago
lbh_qys
2 hours ago
J
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The line is a common tangent
Rijul saini   3
N 2 hours ago by pingupignu
Source: India IMOTC 2025 Day 4 Problem 3
Let $ABCD$ be a cyclic quadrilateral with circumcentre $O$ and circumcircle $\Gamma$. Let $T$ be the intersection of tangents at $B$ and $C$ to $\Gamma$. Let $\omega$ be the circumcircle of triangle $TBC$ and let $M(\neq T)$, $N(\neq T)$ denote the second intersections of $TA,TD$ with $\omega$ respectively. Let $AD$ and $BC$ intersect at $E$ and $\Omega$ be the circumcircle of triangle $EMN$. If $AD$ intersects $\Omega$ again at $X \neq E$, prove that the line tangent to $\Omega$ at $X$ is also tangent to $\omega$.

Proposed by Malay Mahajan and Siddharth Choppara
3 replies
Rijul saini
Yesterday at 6:47 PM
pingupignu
2 hours ago
J
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One of P or Q lies on circle
Rijul saini   6
N 2 hours ago by ZVFrozel
Source: LMAO 2025 Day 1 Problem 3
Let $ABC$ be an acute triangle with orthocenter $H$. Let $M$ be the midpoint of $BC$, and $K$ be the intersection of the tangents from $B$ and $C$ to the circumcircle of $ABC$. Denote by $\Omega$ the circle centered at $H$ and tangent to line $AM$.

Suppose $AK$ intersects $\Omega$ at two distinct points $X$, $Y$.
Lines $BX$ and $CY$ meet at $P$, while lines $BY$ and $CX$ meet at $Q$. Prove that either $P$ or $Q$ lies on $\Omega$.

Proposed by MV Adhitya, Archit Manas and Arnav Nanal
6 replies
Rijul saini
Yesterday at 6:59 PM
ZVFrozel
2 hours ago
J
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Solutions to Challenging Problems in Middle School Math by Ertan Kay
BlisterBud7S   1
N 3 hours ago by Gavin_Deng
Hi,

I have purchased this book, however cannot locate a copy of their solutions manual. Could someone please help wit this?
1 reply
BlisterBud7S
Yesterday at 2:22 AM
Gavin_Deng
3 hours ago
J
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J
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How to Prepare for USAJMO?
hrithikguy   12
N Apr 10, 2011 by inquisitivity
This past year I have mainly been prepping AMC and AIME problems to make USAJMO, but now that I have made it, I have no idea what to do. I would like to get a positive score. Any advice?
12 replies
hrithikguy
Apr 8, 2011
inquisitivity
Apr 10, 2011
J
How to Prepare for USAJMO?
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hrithikguy
1791 posts
hrithikguy
#1 h  Apr 8, 2011, 7:19 PM • 2 Y
Y by Adventure10, Mango247
This past year I have mainly been prepping AMC and AIME problems to make USAJMO, but now that I have made it, I have no idea what to do. I would like to get a positive score. Any advice?
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-$45!
67 posts
-$45!
#2 h  Apr 8, 2011, 7:29 PM • 3 Y
Y by republicofalvinia, Adventure10, Mango247
A positive score is not as hard as I thought last year. I got a 196 index last year, literally did not prepare at all for USAJMO, and got a 13. Even not solving any problems (I solved 1, did most of another), you can get some partial credit by putting what you have. I was surprised because my goal was to get a positive score as well. We still don't really know what to expect on USAJMO because there has only been one year.

So basically, I don't really know how to prepare. Doing olympiad problems will help, but remember that (I'm assuming it's your first year in USAJMO) it's your first year and you have plenty of time to do better. Feel proud of your accomplishments. I'm not saying you shouldn't prepare, but don't freak out too much.
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e^i_penguin
84 posts
e^i_penguin
#3 h  Apr 8, 2011, 7:55 PM • 6 Y
Y by Adventure10, Mango247, and 4 other users
Getting a decent (15+) score or USAJMO should not be too difficult with good preparation. Here are a few things that might help you score higher:
1. Perfect your proof writing skills. Last year, I solved 4 problems (although there was a minor flaw in 1 of my problems, so I probably only ended up getting 1 point), but only got a score of 17. My guess is that I probably did not write up my proofs well enough and got deducted because the readers could not understand my work.
2. From my experience last year, most of the USAJMO problems were just hard AIME problems, and pretty much dealt with the same topics. Therefore, in addition to using last year's USAJMO problems at practice, try doing #10-15 from each AIME.
3. USAJMO problems are usually pretty straightforward and do not require the depth that USAMO problems do. So, doing USAMO problems to prepare for USAJMO isn't going to help you much (that's what I did last year and it didn't help me much).
4. During the test, don't get nervous if you can't solve a problem and don't get jumpy when time is flying by and you haven't made any progress (at most, if you're stuck on a problem for too long, go to another problem you have not finished yet). Doing so will only prevent you from focusing on the questions.
5. Taking the test should be a fun experience, especially since you don't have any pressure on you (I'm assuming you're not aiming for Red MOP?). Just sit back and enjoy doing the problems.
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hrithikguy
1791 posts
hrithikguy
#4 h  Apr 8, 2011, 8:34 PM • 2 Y
Y by Adventure10, Mango247
Thank you for the excellent post, e^i_penguin. Do you know what MOP colors you can qualify for from USAJMO?
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themorninglighttt
789 posts
themorninglighttt
#5 h  Apr 8, 2011, 8:35 PM • 2 Y
Y by Adventure10, Mango247
hrithikguy wrote:
Thank you for the excellent post, e^i_penguin. Do you know what MOP colors you can qualify for from USAJMO?

I think only the winners make Red.
Or CGMOers.
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e^i_penguin
84 posts
e^i_penguin
#6 h  Apr 8, 2011, 10:47 PM • 2 Y
Y by Adventure10 and 1 other user
themorninglighttt wrote:
hrithikguy wrote:
Thank you for the excellent post, e^i_penguin. Do you know what MOP colors you can qualify for from USAJMO?

I think only the winners make Red.
Or CGMOers.
Yes, JMO'ers can only make Red MOP
They pick the 12 winners to make Red and then there are 12 honorable mentions (who unfortunately don't make anything).
However, I will say that it is pretty difficult to get top 12 based on last year and if you cannot consistently solve most of #15s on AIME, it requires a bit of luck. Last year's cutoff was 35, which is 5 questions completely correct (or 4 mostly correct with some partial credit on the remaining 2). So basically, if a problem shows up that you don't know how to do, then you're pretty much already out.
However, that is only the cutoff from last year and is no indication for this year.
If you want to see what the difficulty of the JMO problems, here is the AOPSWiki link: USAJMO Problems/Solutions (if you have not already started looking at them)
Hope that helps.
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abcak
481 posts
abcak
#7 h  Apr 9, 2011, 2:01 AM • 3 Y
Y by Adventure10, Mango247, and 1 other user
You don't really have much time though...your in my Oly Geo class, and that will hopefully overkill the geo problems. Otherwise, proof writing is what I'd recommend if you've had no experiance w/ it.
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inquisitivity
173 posts
inquisitivity
#8 h  Apr 9, 2011, 12:35 PM • 2 Y
Y by Adventure10 and 1 other user
I would also strongly recommend that you work on proof-writing. For example, last year I was only able to get the decent score of 21 because I spent a lot of time writing mathematically detailed proofs (I only solved three problems, with no partial credit on the other ones.).

As far as problem-solving practice, I would recommend that you try working on solving (and writing proofs for) problems #13-#15 on AIMEs. If you find success with that and it becomes fairly easy, I would suggest you solve previous 1's and 4's on USAMOs, particularly for easier tests.
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hrithikguy
1791 posts
hrithikguy
#9 h  Apr 9, 2011, 6:38 PM • 2 Y
Y by Adventure10 and 1 other user
Thank you for the advice, everyone! As an additional question,

Does anyone know of any books for bridging the gap between AMC/AIME type math and olympiad level math?
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exmath89
2572 posts
exmath89
#10 h  Apr 9, 2011, 11:51 PM • 2 Y
Y by Adventure10, Mango247
inquisitivity wrote:
I would also strongly recommend that you work on proof-writing.

Are there any books that can help improve proof-writing for the JMO?
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cwein3
148 posts
cwein3
#11 h  Apr 10, 2011, 12:07 AM • 2 Y
Y by Adventure10, Mango247
Quote:
Does anyone know of any books for bridging the gap between AMC/AIME type math and olympiad level math?

Art and Craft of Problem Solving is very good for that. However, despite whatever book you decide to get, I still think doing past USAMO/JMO problems is the best thing you can do. Once you solve a few of them, they'll appear a lot less intimidating in general.
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NuncChaos
781 posts
NuncChaos
#12 h  Apr 10, 2011, 12:56 AM • 2 Y
Y by Adventure10, Mango247
Personally, since USAMO/JMO problems are limited... in the beginning I recommend Irish MO, CMO, and then APMO before you start to use up IMO, USAMO, and JMO problems...
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inquisitivity
173 posts
inquisitivity
#13 h  Apr 10, 2011, 11:28 PM • 2 Y
Y by Adventure10 and 1 other user
exmath89 wrote:
Are there any books that can help improve proof-writing for the JMO?

I don't know of any books that specifically work to improve proof-writing skills, but, if you haven't already looked at it, AoPS has an article on the topic that might help (http://www.artofproblemsolving.com/Resources/articles.php?page=howtowrite).
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